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A classification of trees and applications of topology and graph theory to neurosurgery

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A classification of trees and applications of topology and graph theory to neurosurgery
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Yi, Tae-Il, 1962-
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viii, 105 leaves : ill. ; 29 cm.

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Algorithms ( jstor )
Brain neoplasms ( jstor )
Dosage ( jstor )
Geometric centers ( jstor )
Integers ( jstor )
Mathematical sequences ( jstor )
Mathematics ( jstor )
Radiosurgery ( jstor )
Tumors ( jstor )
Vertices ( jstor )
Dissertations, Academic -- Mathematics -- UF ( lcsh )
Mathematics thesis, Ph.D ( lcsh )
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theses ( marcgt )
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Thesis:
Thesis (Ph.D.)--University of Florida, 2000.
Bibliography:
Includes bibliographical references.
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Printout.
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Vita.
Statement of Responsibility:
by Tae-Il Yi.

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University of Florida
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A CLASSIFICATION OF TREES AND
APPLICATIONS OF TOPOLOGY AND GRAPH THEORY TO
NEUROSURGERY











By

TAE-IL YI


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


2000





























Copyright 2000


by


TAE-IL YI






















TO

My parents, Beom-Young & Bong-Wol,

who show me how to love,

My wife, Moon-Sil,

who shows me what is love,

and

My children, David & Peter,

who are the reasons why I love














ACKNOWLEDGEMENTS


I would like to express my sincere appreciation to the members of my supervi-

sory committee, Dr. Beverly Brechner, Dr. Philip Boyland, Dr. James Keesling, Dr.

Li-Chien Shen, and Dr. Mary Kantowski, for their patience, knowledge, concern, and

especially for showing me how to study and teach. I would especially like to thank

my advisor, Dr. Beverly Brechner, who accepted me and gave me a chance to work

with her. I also want to express my appreciation for all her effort and all that she has

done for me, not only in my major field, but also in my life. I would like to thank

Matthew Harvey who brought me some wonderful ideas and wrote the Mathematica

programs. I would like to thank Dr. Francis Bova who brought me these questions

which were 'new world' to me. I would like to thank Dr. Meeks for helpful conver-

sations, and Dr. Thomas Wagner who gave me the information about neurosurgery

that I did not know before. I would like to thank Dr. Yunmei Chen who gave me

inspiration. I would like to thank Dr. Miklos Bona who showed me some insights of

combinatorics. Finally, I would like to thank the Department of Mathematics that

allowed me to be here and meet all these wonderful people.














TABLE OF CONTENTS




ACKNOWLEDGEMENTS ............................ iv

ABSTRACT ... .. .. .. ... .. .. .. .. .. .. .. .. ... vii

CHAPTERS

1 INTRODUCTION .............................. 1

2 AN AUTOMATED SPHERE PACKING PLAN FOR BRAIN TL'UMORS 5

2.1 Introduction . .. . . 5
2.2 Linac . . . . .. 5
2.3 Procedure of Radiosurgery ........................ 6
2.4 Making a Sphere by Arcs of Beams .................... 10
2.5 Sphere Packing Plan ......................... 10
2.6 Shelling Algorithm .......................... 15

3 A CLASSIFICATION OF UNLABELED TREES ............ 25

3.1 Introduction .. . . . .. 25
3.2 Definitions .. .. .. .. .. .. .. ... .. .. .. .. ... 26
3.3 Vertex Sequence ........................... 28
3.4 Degree Sequence ........................... 36
3.5 Perfect Sequence and Main Theorem .................. 44
3.6 Obtaining Perfect Sequences . . ... .. 45
3.7 Extended Partitions and the Algorithm . .... ..53

4 TREES AS INVARIANTS FOR BRAIN TUMOR SHAPES ...... 59

4.1 Introduction . . . ... .. 59
4.2 Definitions and Some Properties . ... .. 60
4.3 Graphs for Sphere Packing Plans . ... .. 64
4.4 Order in the Vertex Set . . ... .. 65
4.5 Maximal Tree from a Graph . . ... .. 71
4.6 Graph from a Maximal Tree . . ... .. 81
4.7 Algorithm for Our Maximal Tree . ... .. 81








APPENDICES

A SHELLING PROGRAM FOR SPHERE PACKING .......... 82

A.1 Program for Data Reading ....................... 82
A.2 Program for Center Searching ...................... 83

B PROGRAM FOR THE MAXIMAL TREE OF A GRAPH ...... 91

C PROGRAM FOR CLASSIFYING TREES BY SEQUENCES ..... 93
C.1 Program .. .. .. ... .. .. .. ... .. .. .... 93
C.2 Some Program Results ....................... 97

D LABELED GRAPHS AND TREES ...................... 102

D.1 Number of Labeled Graphs and Labeled Trees ......... 102
D.2 Sketch of the Construction of a Priifer-code ............. 102
D.3 Reconstruction of a Tree from a Priifer-code ........... 103

REFERENCES ................................... 105

BIOGRAPHICAL SKETCH ............................ 105














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy




A CLASSIFICATION OF TREES AND
APPLICATIONS OF TOPOLOGY AND GRAPH THEORY TO
NEUROSURGERY

By

Tae-Il Yi

August 2000


Chairman: Dr. Beverly L. Brechner
Major Department: Mathematics

The work of this dissertation was motivated by our work with a medical group

at the Brain Institute at the University of Florida. This group was led by Dr. Francis

Bova, who raised the following questions:

(1) Can you automate the time-consuming sphere packing process for our "multiple

isocenters method of stereographic radiosurgery" for brain tumors? (This surgery is

noninvasive, radiation surgery to destroy brain tumors.)

(2) Can you classify the shapes of brain tumors?

In this dissertation, we construct some algorithms and programs to (partially)

answer these questions. Our major (joint) results are described below.

Our first result is an algorithm and a program that produce a unique sphere

packing plan for any given brain tumor shape, filling the shape with different size

spheres in a unique way. This process automates the 'sphere packing' part of the








surgery plan. Additional code by the medical group was necessary to allow for physi-

cian input in certain critical cases. It was then possible for the medical group to

incorporate this into their surgery plan, semi-automating it, and significantly reduc-

ing the surgeons' surgery planning time.

Our second result is a complete classification of unlabeled n-trees. We assign

to each n-tree, a sequence of n nonnegative integers satisfying certain properties; and

from each such sequence, we can recover the n-tree assigned to it. In fact, we present

both an algorithm and code to produce not only the sequences, but also pictures of

the corresponding trees.

Our final result is a first attempt at a topological classification of tumor shapes.

We first use our sphere packing plan to construct a graph representing the tumor.

We then have an algorithm and program which uniquely select a maximal tree in the

graph. Thus we can assign a unique tree to each tumor shape. However, different

graphs may produce the same tree. Our tree classification allows us to compare

two tumor shapes very easily, by just looking at their corresponding sequences. In

particular, different sequences come from different sphere packing plans, but the

converse does not hold.














CHAPTER 1
INTRODUCTION

The work of this dissertation was motivated by two questions raised by Dr.

Francis Bova, of the Brain Institute, Shands Hospital, and the Department of Neuro-

surgery, all at the University of Florida. It turns out the different-sounding questions

are very much related and are tied together naturally in this thesis. In fact, our

results for the second question depend upon our results for the first question. Dr.

Bova led a medical group, which joined with our mathematics group, for a number

of discussions over a two year period. This resulted, for us, in both a joint medical

paper [11] and in this dissertation. We also want to reference the dissertation [10] of

Thomas Wagner, Dr. Bova's medical physics student, who worked with us.

We expect to have three mathematics publications from this dissertation-from

Chapters 2, 3, and 4, respectively. The paper resulting from Chapter 2 will be joint

with Professor Beverly Brechner, Professor Yunmei Chen, and Dr. Thomas Wagner.

The papers resulting from Chapters 3 and 4 will be joint with Matthew Harvey, an

undergraduate student working with Professor James Keesling on his honors thesis.

Further, we anticipate an additional paper, to be joint with Yunmei Chen and her

student, Dr. Stacey Chastain. This anticipated work will be an analysis of the tumor

shapes of approximately 1000 interesting tumor radiation surgeries done by Bova and

his colleagues at the University of Florida.

Dr. Bova's questions were the following:

1. Is it possible to automate the medical group's "multiple isocenters method of

stereographic radiosurgery" to destroy brain tumors?

2. Is it possible to classify brain tumor shapes?








In this dissertation, we develop some algorithms and programs related to these

questions. In stereographic radiosurgery, which is noninvasive surgery, radiation is de-

livered by means of collimators-cylindrical metal tubes, with cylindrical holes through

them, through which the radiation flows. The goal of stereotactic radiosurgery for a

brain tumor is to deliver the desired dosage to the target, and only the target. This

is not possible in reality. So they do the next best thing, which is to deliver enough

dosage to the target, to avoid as much normal tissue as possible, and to deliver as

little radiation as possible to whatever normal tissue must be affected. There are two

additional important criteria-dose homogeneity and dose conformality. That is, we

do not want 'hot spots,' which have been experimentally determined to cause com-

plications; and we do want rapid falloff of dose levels outside the actual tumor. One

of several such radiation surgery methods is called the 'Multiple Isocenter Method.'

This involves filling the tumor image with spheres of different sizes, until the image is

best filled up. In this dissertation, we call such a plan, a 'Sphere Packing Treatment

Plan.'

In Chapter 2, we develop both an algorithm, which is called a 'shelling process'

(or grassfiree process' in medical terminology), and a C-H-+ program to identify the

centers and sizes of spheres for this sphere packing treatment plan. For any given

brain tumor, our program gives us a unique sphere packing plan. This sphere packing

plan can be represented numerically as a list of parameters of the spheres, namely

the centers and radii of the spheres. Based on this algorithm, the medical group

developed a proprietary program for an automated sphere packing treatment plan,

including dosage distribution, and other things (see Wagner et al.[ll]). Since this

program always produces the same plan for the same tumor, it can be used as a

'benchmark' against which to compare other treatment plans.

In Chapter 3, we obtain a code to denote an unlabeled tree by a unique,

ordered, nonnegative integer sequence satisfying certain properties. We call such








sequences, perfect sequences. That is, we obtain a complete classification of unlabeled

n-trees by means of the perfect sequences of n nonnegative integers. Thus we have our

main theorem of this chapter, the Tree Classification Theorem, (Theorem 3.5.5): For

any positive integer n, let T(n) be the set of unlabeled n-trees and P(n) the set of

perfect sequences of n-trees. Then there is a one-to-one correspondence between T(n)

and P(n). We show how to assign a perfect sequence to a given tree, and conversely,

how to reconstruct a tree from a given perfect sequence. We also include an algorithm

and a Mathematica program to produce not only all the perfect n-sequences, for any

given positive integer n > 3, but also pictures of the corresponding trees. By means of

this program, we get our classification of unlabeled n-trees and also the total number

of n-trees.

In Chapter 4, we make a first attempt at a topological shape classification of

tumors-actually of the sphere packing plan associated with the tumor shape. We first

assign to each tumor a unique graph, and from that, we determine a unique tree. We

obtain the graph by assigning a vertex for each sphere, and an edge between each pair

of vertices representing adjacent spheres. This produces a unique connected graph

corresponding to each sphere packing plan. Then, by using the notion of cutvertex, we

give an order to the vertex set. We then use this order to build a unique maximal tree

contained in the graph. Next, we develop an algorithm and a Mathematica program

to assign a unique tree to each sphere packing plan. This program produces a list of

edges of the tree, making it clear which edges are incident.

Using our results of Chapter 3, we can now assign exactly one perfect sequence

to each tumor. However, it may happen that more than one tumor shape (or graph)

gives rise to the same tree, and therefore to the same perfect sequence. Thus, we do

not yet have a real classification of tumor shapes. However, if two perfect sequences

are distinct, then their corresponding trees, and therefore their respective correspond-

ing graphs and sphere packing plans, are also distinct. Therefore we may consider





4


their shapes to be distinct. Thus, we have our main result of Chapter 4, Theorem
4.5.6: The perfect sequences are invariants of the shapes of arbitrary brain tumors.

That is, if two brain tumors are represented by distinct perfect sequences, then their
corresponding trees are not isomorphic, and therefore their respective graphs and
sphere packing plans are not isomorphic. Thus, we may consider their shapes to be
distinct.
We refer the reader to Douglas[l], Gross[3], and Harary[5] for graph theory.
We also refer the reader to Hall[4], Riordan[7] and Stanley[9] for combinatorics.














CHAPTER 2
AN AUTOMATED SPHERE PACKING PLAN FOR BRAIN TUMORS

2.1 Introduction

The goal of stereotactic radiosurgery for a brain tumor is to deliver the desired

dosage to the target, and only the target. This is not possible in reality. So they do

the next best thing, which is to deliver enough dosage to the target, to avoid as much

normal tissue as possible, and to deliver as little radiation as possible to whatever

normal tissue must be affected. There are two additional important criteria-dose

homogeneity and dose conformality. That is, we do not want 'hot spots,' which

have been experimentally determined to cause complications; and we do want rapid

falloff of dose levels outside the actual tumor. One of several such radiation surgery

methods is called the 'Multiple Isocenter Method.' This involves filling the tumor

image with spheres of different sizes, until the image is best filled up. This noninvasive

method of surgery, namely by using radiation, relies on a piece of equipment called the

Linear Accelerator (or simply, Linac). In this chapter, we show how to automate the

construction of a sphere packing plan for the tumor image. Most of the information

in this chapter about treatment of a brain tumor is taken from Friedman et al.[2].

See [2] for further reference.

2.2 Linac

According to Friedman et al.[2], the linear accelerator is a complex machine

capable of producing X-rays. A large amount of energy is generated by the power

supply, which then powers the filament shown. This causes electrons to be emitted

by the filament, which are in turn accelerated to higher energies using a (microwave)








wave guide. The electrons are then changed in direction by the magnet so that they

impact on a heavy metal alloy target. This results in X-ray production that can then

be collimated or shaped by both primary and secondary collimators within the linear

accelerator head. This beam is further collimated for radiosurgery by the tertiary

radiosurgery collimator. (See Figure 2-1 and Figure 2-2.)

The Linac is mounted on a rotating gantry such that the beam has a center of

rotation about 1.5m above the floor. Usually, the isocenter accuracy is defined within

a 2mm sphere. Because stereotactic radiosurgery depends on optimized accuracy, an

improved system was designed at the University of Florida, by adding a set of bearings

to the stereotactic collimator system and under the patient table. As a result, this

new system achieves mechanical accuracy within 0.2mm0.1mm for defining the

treatment isocenter of beam delivery.

The tertiary collimators are generally circular and allow improved centering

of the treatment beam. The sizes of these collimators are from 5 to 40mm in 2- to

5-mm increments.

2.3 Procedure of Radiosurgery

The procedure of radiosurgery is separated into 5 steps as follows.

(1) Head ring attachment

(2) CT scan

(3) Image Processing

(4) Planning

(5) Treatment delivery

For more detailed information, refer to Friedman et al.[2].

(1) Head ring attachment

The stereotactic radiosurgery treatment requires attachment of a stereotactic

head ring. This ring provides accurate information about the position and the shape



















Heavy alloy target

Primary (Internal)
collimators
Flattening filter

Secondary (Adjustable)
colimators
Tertiary (Radiosurgery)
collimators
Photon beam for
radiourgery


Floor stand with treatment arm above


Figure 2-1. System of Linac (Friedman et al.[2])






























Figure 2-2. Operation of Linac (Friedman et al.[2])








of brain tumors from the computer tomography (CT). This ring also helps to insure

that the treatment target is accurately placed at the precise isocenter of the Linac.

(2) CT scan

After attaching the head ring, a series of cut filming is taken. Eighty to 150

CT slices are taken per patient. For radiosurgical treatment planning, getting the

accurate size and shape of the target is very important. Underestimation of the target

size may result in treatment failure. Overestimation of size results in the inclusion of

normal brain tissue within the treatment volume. Misrepresentation of an irregular

target shape may lead to radiation damage of normal brain tissue. The accuracy

of CT scanning depends on the size of the pixels that make up the scan image.

The lateral and AP accuracy of scanning is approximately 0.6mm. But the axial

dimension is determined by slice thickness of the CT scan, which is 1mm in general.

The physicians use the so-called Brown-Roberts-Wells coordinates-or, simply BRW

coordinates-which are called lateral (Lat), anterior-posterior (AP), and axial (Ax)

coordinates. These are the x, y, and z coordinates, respectively. The direction of

the lateral coordinate is from the left hand side ear to the right hand side ear of the

patient. So the lateral view (or sagittal) is similar to the side view of the head, that

is, views of slices along the lateral axis. This is the same as yz-plane views. The

direction of the AP coordinate is from the back side of the head to the face. So the

AP view (or coronal) is similar with the front view of the head, that is, views of slices

along the AP axis. This is the same as xz-plane views. The direction of the axial (or

vertical) coordinate is from the neck to the top of the head. So the axial view (or

trans-axial) is similar to the sky view of the head, that is, views of slices along the

axial axis. This is the same as xy-plane views. (See Figure 2-4.)

(3) Image Processing

An MRI scan has been done the day before surgery. Using a mouselike device,

the CT and MRI images are transferred to the computer screen and fused. Then,








using geometric equations, the computer determines the lateral, AP, and vertical

stereotactic coordinates of each point in each CT slice.

(4) Planning

Physicians use a mouse and their own judgment to 'eyeball' the location and

the size of spheres for the treatment plan.

(5) Treatment delivery

The geometric sphere packing plan, developed as in (4), is combined with a

proprietary dose planning program, and the resulting treatment plan is delivered by

the Linac.

2.4 Making a Sphere by Arcs of Beams

By varying the angle of the gantry and the angle of the table, one can deliver

a radiation beam to the target from any angle within the range of the rotation. The

shape of the common intersection of an arc of beams passing through one isocenter is

a sphere. The neurosurgeons deliver a series of arcs (usually 5 or 9 arcs) to produce

a single isocentered sphere shape. For an ellipsoidal target, they use fewer numbers

of arcs to make a single isocentered ellipsoid shape. (See Figure 2-3, 2-4 and 2-5.)

So, if the target shape is very close to a sphere or an ellipsoid, then the

treatment plan is relatively easy compared to an irregularly shaped target. In that

case, we need to create a geometric treatment plan.

2.5 Sphere Packing Plan

As seen in the previous section, the physicians know how to irradiate-to-
(eventually)destroy tumors which are shaped like spheres or ellipsoids. For a non-

spherical shape of tumor, they try to fill the target with several spheres of different

sizes. This is called the 'sphere packing' treatment plan.

After finding a sphere packing plan, they treat each sphere separately as de-

scribed in the previous section. So, multiple isocenter radiosurgery planning includes
































Figure 2-3. One isocenter plan with five arcs of radiation (Wagner et al.[ll])













(270')


(310*)


(330")

(350) I


(501


(30')

A (lo)


I H G FED C BA


Figure 2-4. One isocenter plan with nine arcs of radiation (Friedman et al.[2])





13




A F
,,.. ,,"" : ,-_ _._ .... **\ '*,f
%1% Ii% l II I
.' -./ .-,,,, i, ,,,
> /-" "l '. *./,

.,TARGET : _
I,





B

No
TARGET .,

Figure 2-5. Plan for a sphere shape and an ellipsoidal shape (Friedman et al.[2])
"'- i \ ': "' "* "" "* "',' -

.-.., 2::_'V :2 *

/ ..' : "*
Figure 2-5. Plan for a sphere shape and an ellipsoidal shape (Friedman et al.[2])








the problem of determining the best sphere packing arrangement with which to fill

the target volume. General methods for this treatment plan are iteratively based,

dosimetrically driven algorithms. But these methods require many computations in

order to compute a radiosurgical plan dose distribution, and then to evaluate the

quality of the dose distribution. So geometrically based radiosurgery optimization

has been suggested as a possible alternative means.

However the method the physicians choose relies on human decisions and

experience. Thus, for the same target, different surgeons may produce different plans.

Even the same surgeon, doing the plan twice for the same target, may produce

different plans. And the planning takes a long time, especially for a complicated

target which needs more than 10 spheres. It might take as much as two hours of

planning for a difficult case which needs about 20 spheres. During that time, the

patient has to wait with the head ring attached to his or her head. And most

importantly, even after spending the time to make a plan, many physicians without

sufficient experience, are not sure if the plan is a 'good' one.

Therefore, we provide an automated sphere packing method for the treatment

plan (see Wagner et al.[11]). This method shows potential to significantly aid the

planning of difficult multiple isocenter cases. Based on tests with irregularly shaped

phantom targets and with a representative sampling of clinical example cases, the

method demonstrates the ability to generate radiosurgery plans comparable to, or

of better quality than, multiple isocenter Linac radiosurgery plans found in other

literature. At the same time, this program always produces the same treatment plan

for the same tumor shape. So it can be used as a 'benchmark' to compare with other

plans for the given tumor shape. Moreover, this program provides the treatment

plan in a relatively short time. For a very difficult case which needed more than 18

spheres, this program took less than 3 minutes instead of more than the 1.5 hours








which were needed when the physicians created the plan using traditional methods.

The algorithm of this program is shown in Figure 2-6.

In the following section, we explain the 'shelling algorithm' which we used in

this program to get the centers and sizes of spheres for the sphere packing plan.

2.6 Shelling Algorithm

The shelling procedure is best illustrated in Figure 2-7 to 2-16. The major

steps of this shelling procedure are as follows.

(1) Transferring the data of the boundary of the target

From the CT scan, each point which is identified from the target contour data

file can be mapped into a three dimensional array, since each CT scan shows the

trans-axial view for a specific axial coordinate value. In the three dimensional array,

we assign the lateral-, AP- and axial-coordinates to be x-, y- and z-axes, respectively.

For each individual case, there are minimum and maximum values for x, y and z-

axes. We assume the minimum value is always one. We get this information from

the Program for Data Reading. (See Appendix A.1.)

Let the target be minimally enclosed in an x-range beginning with one and
ending with M,, a y-range beginning with one and ending with My, and a z-range

beginning with one and ending with M,. Now we put 'safe bumper' layers sur-

rounding the target. That is, we enlarge the box containing the target to ET =

[0, M, + 1] x [0, My + 1] x [0, M, + 1]. ET stands for 'enlarged target box.' We assume

that the dimension of each voxel is 1 x 1 x 1. Then we will number the voxels in a lexi-

cographic manner, beginning with 0 and ending with (M.++l)x(My+l)x(M2+1)-1,

giving us a total of (M, + 1) x (My + 1) x (M, + 1) voxels with which we will work.
Now we want to identify the voxels in ET in two different ways: first, by a set

of coordinates (a, b, c), and second, by an integer representing a counting procedure.

Here, a is the left hand endpoint of the x-value of that voxel, b is the left hand








endpoint of the y-value of that voxel, and c is the left hand endpoint of the z-
value of that voxel. The second notation for the voxel (a, b, c) is by some integer
n. The voxels are counted, using a lexicographic ordering beginning with 0 and
ending with (M, + 1) x (My + 1) x (M, + 1) 1 which is denoted by nofdata in
the program (see Appendix A.2). Then the voxel denoted by (a, b, c) is numbered
as c x (M. + 1) x (My + 1) + a x (M. + 1) + b. And it is clear that, for every
0 < i < nofdata, there are unique nonnegative integers, a,b,c, such that i =
c x (M, + 1) x (M. + 1) + a x (My + 1) + b. That is, for every 0 < i < nofdata,
one and only one voxel is assigned. Note that Figure 2-7 to Figure 2-14 show the
same trans-axial view at different stages of the algorithm. Thus these figures show
the data of a fixed axial-coordinate value.
(2) Assign a status value for each voxel
For every 0 < i < nofdata, the status value for the i-th voxel is denoted
by cc[i]. That is, we denote the status value for the voxel with coordinates (a, b, c),
by cc[i] where i = c x (M, + 1) x (M. + 1) + a x (M. + 1) + b. The following is
the procedure to assign status values. First of all, let cc[i] = -3 for every 0 <
i < nofdata. Then we assign the number, -1, as the status value for each voxel
corresponding to any of the target contour data. Next, for every 0 < j < M,, we
assign cc[j x (M, + 1) x (My + 1)] = 0, because the j x (M, + 1) x (My + 1)-th voxel
is the voxel at the left lower corner for every j-th trans-axial view; that is, the voxel
belongs to the healthy tissue. (See Figure 2-8.)
For each voxel with cc[i] = -3, we define cc[i] = 0 if any of cc[i 1], cc[i +
1], cc[i (My + 1)], cc[i + (My + 1)] is 0. Note that this method produces zeros for
every voxel in the safe bumper layers. (See Figure 2-9.)
(3) Shelling the target voxels
We proceed by mathematical induction on j, for

0 < j < x min{Mn My, Mz} .
12








For i = 0 to (M, + 1) x (My + 1) x (Mz + 1) 1, if the i-th voxel is assigned -1
or -3 for its status value, and if any of cc[i 1], cc[i + 1], cc[i (M, + 1)], cc[i +
(My + 1)], cc[i (M, + 1) x (My + 1)], cc[i + (M. + 1) x (My + 1)] is equal to j,
then we assign a new status value for the i-th voxel as c[i] = j + 1. Note that Figure
2-7 to Figure 2-14 show only a two-dimensional figure. But this procedure is held
in a three-dimensional array. This process is repeated until the deepest lying voxels
have been identified. The deepest lying voxel in the entire target volume would be
the best location for an isocenter. And at the same time, the status value assigned to
the deepest voxel indicates the radius of the sphere, which is one less than the status
value of the voxel. This completes our induction. (See Figure 2-10.)
(4) Removing the largest sphere
At this moment, we might have several different voxels which have the same
largest status value. So we choose the smallest numbered voxel as the center of the
first largest sphere to be removed from the target. But in the program in Wagner et
al.[11], we choose the most effective voxel to be the center of the first largest sphere,
by using the 'score function' which was developed by Tom Wagner. We think of
the resulting sphere in the tumor as 'removed,' and therefore we can assign the new
status value zero for these voxels in the sphere. To determine which voxels form the
sphere that is removed, we do the following: Let R = voxel at the center of sphere
to be removed. Let r = status value of R. Then we remove all voxels S with the
property that distance(center of R, center of S) < r 1. (See Figures 2-11 and 2-12.)
Now, we proceed by mathematical induction on j, for

0 < j K -x min{Mn My, Mz} .

For i = 0 to (Mr + 1) x (My + 1) x (M, + 1) 1, if the i-th voxel has status value
> j, we reassign a new status value as follows: if any of cc[i 1], cc[i + 1], cc[i -
(My+ 1)], cc[i+(My+l)], cc[i-(M+l)x(My+l 1)], cc[i+(M.+1)x(My+l)] is
equal to j, then we assign a new status value for the i-th voxel as c[i] = j + 1. This








process is repeated until the deepest lying voxels have been identified. (See Figures

2-13 to 2-15.)

And we repeat step (4) until we get either 0 or 1 for the status value of each

voxel in ET, the enlarged target box. Because the smallest size of the diameter of

collimators is 5mm, we finish our program when the status values of all the voxels are

zeros and/or ones. (See Figure 2-16.) The remaining part will be taken care of by

the 'score function' (see Wagner et al.[11]). After finishing this procedure, we have

the spheres filling the target. In Figure 2-17, we see the sphere packing plan within

a target by this method.

For further detailed information about the score function and the automated

sphere packing plan, we refer the reader to Wagner et al.[11] and Wagner[10].


STOP


Figure 2-6. Sphere packing algorithm block diagram (Wagner et al.[ll])












0 *0 0







00
.0
0 0





0
0






0 5 10 15 20 -

Figure 2-7. Transferring the data of the boundary of the target




-3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3
-3 -3 -3 3-3 -3 -3 -1 -1 -1 -1 -1 -1 -1 -3 -3 -3 -31-3 -3 -3 -3 -3
-3 -3 -3 -1 -1 -1 -1 -1 -3 -3 3-3 -3 -1 1-1 -1 -1 -1 3-3 -3 -3 -3
-3-33-1,-3 -3 3-3 3 3 -3-3-3-3 3-1 -1 -3 1-1 -3 -3 -3 -3.
-3 -1 -1 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -1 -3 -3 -3
-3 -1 -3 -3 -3 3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -1 -1 -3 -3
-3 -1 -3 -3 3-3 -3 -3 -3 -3 -3 -3 -3 -3 3-3 -3 -3--3 -33 -1 -1 -3
-3 -1 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -1 -1 -3
-3 -1 -1 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -1 -3 -3
-3 -3 -1 -3 3 -3-3 3 3-3-3-3 3 3 -3 -3 -3 -3 -1 -1 -1 -3 -3
-3 -3 -1 -1 -1 -1 -3 -3 -3 -1 -1 -3 -3 -3 -3 -3 -3 -1 -1 -3 -3 -3 -3 -3
-3 -3 3-3 -3 3 -1 -1-1 -1 -3 1-1 -1 -1 1 -1 -3 -3 -3 -3 -3 -3 -3
-3 -3 -3 -3 3-3 -3 -3 -3 -3 -3 -3 -3 -1 1 -3 -3 -3 -3 -3 -3 -3 -3 -3
033 -31-31-3 -3 -3 -3 -3333 -3 -3 -3 -3-3 -3 -3 -3 -3

Figure 2-8. Status values for the boundary and the initial voxel









0
0
0 1
0
0
0
0
0
0v
0 1
0
0
0 1
0 o


o o |o |o |o |ol|ol|ol|o/|o/|o/|ol|o/|o |ol|o |o |o |o o o o o
)1o o o o o10ai o o o 01o o o o o o
001 0 0., 000000000
00 0 0 0 0
0 00 0 0


JO 0000 o tooolololololo ofolojololo


010


0 0


Figure 2-9. Status values for the normal voxels






















Figure 2-10. Shelling the target voxels


I







0 0 0 0 0 0 0 0 0 0 0 0 0 1 -000 0 0 0 0 0
0 0 U 0 0 0 0 1 1 1 1 1 1] 1 0 0 J 0 0 0 l0 u

S0 1 2 2 2 3 3 3 3 3 11 1 1 1 0 0 0 0
1i 3 3 3 13 4 4 4 4 4

.1 4 4 4 4 5 5 5 5 4
0-i-1 4 5 5 5 4 4 5 5 A 1 0.
0 1 1 'l33 4 4 4 3 3 4 4 '1 0 t
0 0 1 1-. 2 2 3 3 3 2 2 3 31 1 0
) I,.) 1 2 21 2 1 1 2 0 0 () oo )
0 0 0 0 0 11 11 1 0 1 1 1 1 1 1 0 0 0 0 0 0




0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0I 0
o0 0 0 0 0 0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0 0 0
Figure 2-11. Choosing the largest sphere



0 0 0 0 0 0 0 0 0 0 ) 0 000
0 0 0 0o III1 0 0 0 0
o I 1 0 1) 0 1) 0 0 0 0 0 1 I 0 'i


O 0 1 .2 l0 0 0 0 0 0 0 0
"2..,0 0 () 0 0 0 0 ..


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 oi
I- .. 1 0 o )u0 0,, 0 o1o
c) i"... 0 01 0 o 0) 0 0 0 0 0 1 ,i 1 1 o

0_ 1 1- 0 0 0 0 0 0 02 1 0 0 .
() 0 0 (.2. 0 0 111 0 10 t) 0 1 1 1 11 0 ) 0

0 1) 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 () 1
0 00 00 000 000 0 0 0 0 0 0 0 0


Figure 2-12. Removing the largest sphere







0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-
0 0 0 0 u 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 0 0 0 0 0 0 1 1 1 11 1 0 0 0 0 j
r0 0 1 1 0 the1 0 r T I I
0 11 0 0 0 u 0 U1) 00 1 1 0 10
U .. .~ I0U~U lT
0 1 1 0 0 0 0 0 0 0 0 0 0 0 1
) To o 1 o o o ( o o o o I T 11
0 1 L-T o 1 0 0 0 o .-) 0 0 0 0 1 O 2 .. 42 1 0 0)
0 1~ 11O0O O O U 1 1 1) 1 1 1 (

U 0 1 1 1 0 o o o o 01 10 00 0 2 41 1OU 1 U 0 1 00o
0 0 1 1 0 0 01 0 0 0 0 0 1 2 1 0000 000
0 0 1 1 01 0 10 0 0 0 0 0 1 0 lo 0 01, o 0 0 0

0 0 0 000 0 0 0 0 01010 0 0 01 0 0 0 0 0 0 0
Figure 2-13. Shelling the remaining target voxels

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0- 0

0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0






0 1.&.1(J 00000_0 U1 '.)11 0 1 23 3 0.Z 1 0 0
0 0 0 0 0 ) 0 1 1 0 1 1 1 0 0 0 0
oI 1ooooooo ,0 o1O0ooO



Fi ur o o o o o o o o o s ofo t r emini o a
o I 1 0 0 0 0 0 0 0 0 o 0 0 142 3arge 1 1t
Q I : 1 0 0 0 0 0 0 0(' ) 0 1l 4 A A 1 1 1,0
I 1 0 0 0 0 0 0 u 1.) 0 1l 3 2 1-. 0 0
0 0 1 1 0 0 0 10- 0 0 0 0 0 1 2 3 3 212 1 1 1 0 0
0 0 1 1 1 0 0 10 0 0 0 0 1 2. 2 1 0 10 0 0 0

0 00 0 0 0 0 0 0 0 0 0 0 1 1 0'L 0] 10 i 0 ) 0 10 0
o 0 0 0 0 0 0 0 0 0 0 0 0 0 0~ 0 0 0 0 00 0
Figure 2-14. Choosing the largest sphere of the remaining target









0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00000 0 0 0
0( 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 o 0 0 (
0 0 0 1 1 0 0 0 0 0 o I I IJI I 1 JJ0-1 -0 0 0 (1 0
0 0 ) 1 1 0 C0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 '
0 1 |1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0
0 1 1 0 0 0 0 00 0 O l0 o1 1 01 o
0 1 1 0 0 0 0 0 0 0 0) 0 0 0 0 0-0 0 01 1 1 0
0 1 1: 0 0 0 0 0 ( 0 0 1 0 0 0 0 1 2 1 1 0
I 1 .2 u 0 0 o 0 0 0 C ) 0 0 0 0 0 0 0 0 0 1 1 ) 0
0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0
0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0o 0 0 o0 0
0 0 0 0 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 0 o 0
0 0 0 0 0 0 0 01 0 1 l 0 0 o 00 1) 0 iJ)
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

Figure 2-15. Removing the second sphere



0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0
0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 (.) 0 1 0 1 1 0 1 1 0 0 0)()
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 1 1 0

0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0
0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 00 0 0 0 00 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0

Figure 2-16. Removing all the spheres












patB.wol






.35---
-40.
i '-- c














-1-"" ,-.^ -' .... ," ...,,
-20 2






100


-20 Lot
AP

Patient A. 12 8cc mengloma







.3)5

-306




i 0i- ^ .....
.40 "





.30
2 -20
10 0 -10
.1

-2D Lat
AP

Figure 2-17. A sphere packing plan for a brain tumor (Wagner et al.[ll])














CHAPTER 3
A CLASSIFICATION OF UNLABELED TREES

3.1 Introduction

In this chapter, we obtain a code to denote an unlabeled tree. By means

of this code, we classify unlabeled n-trees. One may take an unlabeled n-tree to

be the geometric realization of a connected, finite, simplicial complex without any

simple closed curves. In particular, we call a tree an n-tree if and only if it is a tree

with n vertices (that is, (n 1) edges). Our code assigns a unique, ordered, 'perfect

sequence', pf(T) = (di, d2,. ., dn), to each unlabeled n-tree, T. And, conversely,

given an ordered sequence of n integers satisfying certain properties, it is the perfect

sequence of exactly one unlabeled n-tree. Thus, we have our main theorem, the Tree

Classification Theorem (Theorem 3.5.5). We then show how to reconstruct the tree

shape from a given perfect sequence, by hand.

Our work includes an algorithm and a Mathematica program (see Appendix

C.1) that produce a list of all the perfect sequences for all possible n-trees, thus also

producing the number of n-trees, for any given n. For some examples, see Appendix

C.2. Since our algorithm and program are based upon our main theorem, we know

that all possible unlabeled n-trees are produced by the program. Thus the program

gives us, for each positive integer n, a complete list of all unlabeled n-trees.

Given n > 0, with sufficient computer power and time, our program will

list all the unlabeled n-trees, thus telling us how many unlabeled n-trees there are.
However, we do not have a simple formula that tells us how many unlabeled n-trees

there are. This remains an open problem.








The Literature contains work regarding the classification of labeled graphs

and labeled trees. In Lovasz[6], the Priifer-code is used to determine the number of

labeled n-trees (see the Appendix D.2 and D.3). In Sloane[8], Sloane shows a table
for n < 26. However, Sloane references Harary for the method he uses. In Harary[5],
there is a simple formula for the total number of labeled graphs with n vertices (see the

Appendix D.1). There is also a formula for the total number of unlabeled n-trees, but

it requires knowing the number of rooted n-trees, which is calculated inductively. And,
even though Harary[5] contains n-tree shapes up to n = 10, there is no explanation
of how to get the shapes of these trees, while our work does show how to get these
tree shapes from our code.
This chapter is organized as follows:

Section 1: Introduction
Section 2: Definitions
Section 3: Vertex Sequence

Section 4: Degree Sequence
Section 5: Perfect Sequence and Main Theorem
Section 6: Obtaining Perfect Sequences
Section 7: Extended Partition and the Algorithm

3.2 Definitions

Most of the necessary definitions in this section are from Harary[5]. We refer
the reader to Harary[5] for easy reference.

Definition 3.2.1 A graph G = (V(G),E(G)) consists of a vertex set V(G) =

{vi,...,vn} and an edge set E(G) = {el,..., em} C {{fu,v}u, v E V(G)} in which
each edge is an unordered pair of two distinct vertices u and v, and the edge {u, v}
is said to join u and v. For an edge {u,v} we say that u and v are adjacent
vertices and the vertex u and the edge {u, v} are incident with each other, as are








v and {u,v}. From now on, |V(G)I denotes the number of vertices and IE(G)I
denotes the number of edges. For a vertex u E V(G), the vertex set
A(u) = {v E GIv is an adjacent vertex of u}
is called the adjacent set of u. Note that u A(u).

Definition 3.2.2 A walk is an alternating sequence (vo0, ei, v1, e2,... ,ek, Vk) of
vertices and edges s.t. vi-,1 vi for i = 1,2,...,k, and ei = {vi-i,vi} for i =
1,2,..., k. If there is no confusion, a walk can be denoted by a sequence of vertices
only in the path like (Vi, V2,..., Vk) A cycle is a walk (vo, el, vi, e2,...,ek, Vk) with
k > 3 in which v0 = vk is the only vertex repeated. A path is a walk with no
repeated vertex. A (u, v)-path is a path with the first vertex u and the last vertex
v. The length of the (u, v)-path, l(u,v), in a graph is the number of edges in the
(u, v)-path. The distance d(u,v) between two vertices u and v is the length of a
shortest path joining them, if any; otherwise d(u, v) = oo. Note that d(u, u) = 0. A
graph G is connected iff every pair of vertices are joined by a path. A graph having
no cycle is acyclic. Note that an acyclic graph can be disconnected. A forest is an
acyclic graph. A tree is a connected acyclic graph. An n-tree is a tree with n vertices.

Definition 3.2.3 We say that a graph G is isomorphic to a graph H if and only
if there exists a bijection f : V(G) -+ V(H) such that {u, v} E E(G) if and only if
{f(u),f(v)} E E(H).

Definition 3.2.4 An unlabeled tree is an isomorphism class of trees.
We are interested in the total number of unlabeled n-trees and their shapes.
For example, T1 = ({1,2,3},{{1,2},{2,3}}) and T2 = ({1,2,3},{{2,1},{1,3}})
are different trees, but there is an isomorphism f : V(T1) -4 V(T2) by f(1) =
2, f(2) = 1, f(3) = 3. Thus they are the same unlabeled tree, so we want to count
them as one unlabeled 3-tree. From now on, a tree implies an unlabeled tree unless
otherwise specified.








Definition 3.2.5 The degree of a vertex v, defined by d(v), is the sum of the
number of edges incident with v. A vertex v is called an endvertex if d(v) = 1.

We will use the geometric, that is, topological realization of a graph inter-
changeably with 'graph' as defined above.

Proposition 3.2.6 [5] The following statements are equivalent for a graph T:
(1) T is a tree.
(2) Every two vertices of T are joined by a unique path.
(3) T is connected and IV(T)I = IE(T)I + 1.
(4) T is acyclic and IV(T)I = \E(T)I + 1.

Note that there are exactly n vertices and n-1 edges in an n-tree.

Proposition 3.2.7 (Degree sum formula) [5] If G is a graph with the vertex
degrees di,...,dn,, then ZvEv(G) di = 21E(G)I.

Corollary 3.2.8 For an n-Tree T we have

i=1 d(v,) = 2(n 1) = 2E(T)\ = 2(IV(T)I 1)
where vi is a vertex of T for i = 1,2,..., n.

Because a nontrivial tree satisfies EEV(T)d(v) = 21E(T)I = 2(IV(T)I 1),
there are at least two vertices with degree 1. Thus we have the following theorem as
a corollary.

Theorem 3.2.9 [5] Every nontrivial tree has at least two endvertices.

3.3 Vertex Sequence

In this section, we define the vertex sequence of a given n-tree.

Definition 3.3.1 Let T = (V,E) be an n-tree. If d(v) = m, then there are m
components of T {v}. A branch of v (or simply a v-branch) is a subtree which
is the closure of a component of T {v}. We call the set of vertices in a v-branch
a v-branch vertex set. If u E A(v), then the v-branch vertex set containing u is
called the (v,u)-branch vertex set and denoted by B(v,u) (or B(u)). We call the








set B'(v,u) = B(v,u) {v} the (v,u)-subbranch vertex set, which is denoted by
B'(u) if there is no confusion. The subtree consisting of B'(v, u) as the vertex set
and {{s,t} E B'(v,u) x B'(v,u) {s,t} E E(T)} as the edge set is called the (v,u)-
subbranch.

Example 3.3.2 Let T be the 10-tree shown below.

s t u


p q r v w x

Y
'y
Figure 3-1

Since d(v) = 4, there are 4 components of T-{v}. Since A(v)= {r, t,w,y},
we have that
B(v,r) = {v,r,s,q,p}, B'(v,r) = {r,s,q,p}
B(v,t) = {v,t}, B'(v,t) = {t}
B(v,w)= {v,w,u,x}, B'(v,w)= {Iw,u,x}
B(v,y) {v,y}, B(v,y) {y}.

On the other hand, we have that
B(r,v) = {r,v,t,u,w,x,y}.
That is, B(v, r) $ B(r, v). And this is true for any tree except the 2-tree. In fact, for
every n-tree with n > 2, there are at least 3 vertices, namely u, v, and w. Without
loss of generality, let u and v be vertices adjacent to each other, and let the (w, v)-
path contain the vertex u. Then w E B(v,u) and w B(u,v).

Definition 3.3.3 A section of a sequence is a subsequence consisting of consecutive
terms of the sequence.

Definition 3.3.4 Let T = (V, E) be an n-tree with V = {x, x2, ... x}. We define
the set, S, of sequences of the form s = (vI, v2,... v, n), where s is a permutation
of the xi's in V(T), to be those sequences satisfying:








(*) forany vj with 1 where k = max{i {1,2,... ,j3} A(v) {v1, v2, ,v} # 0-
Then S is called the vertex sequence set of T and each element, s, in S is called a
vertex sequence of T.

Definition 3.3.5 For any v E V, we call S, = { E Svi = v} C S the v-vertex
sequence set.

Definition 3.3.4 requires that the vertices in a sequence be listed in a particular
way. The condition (*) ensures that all vertices are listed and that each is listed
exactly once. We can choose any vertex for the first term, vi. Then the condition
(*) says that, for the (i+l)-th term, we have to choose an unlisted adjacent vertex
of vi, if possible. If every adjacent vertex of vi is already used, then look backward
along the sequence until we come to the first vertex which has an unused adjacent
vertex. Then, for the (i+l)-th term, we assign one of the unused adjacent vertices of
that vertex. To get a vertex sequence, we continue this procedure inductively.
Note that every vertex sequence of an n-tree has n entries.

Example 3.3.6 We show how to construct a vertex sequence of a given tree. Let
T be the 7-tree shown below.

f g



a b c d e
Figure 3-2

Since we have freedom to choose the first term, let v, = b. Then v2 E A(vi)-
{vl} = A(b)-{b} = {a,c,f}-{b} = {a,c,f} = A(b) = A(vi). So v2 must beoneof
the elements of A(vi) = A(b) = {a, c, f}. Let v2 = c. Since v3 E A(v2) {v1, v2} =
A(c) {b,c} = {b,d,g} {b,c} = {fd,g}, v3 must be either d or g. Let v3 = g.
Then v4 A(v3) {v,v2, v3}, since A(v3) {v1,v2, v3} = {c} {b,c,g} = 0. On







the other hand, we have that A(v2) {v1,v2, V3} = {b,d,g} {b,c,g} = {d} 5 0.
Therefore we have to choose d as v4. Then v5 = e since A(v4) {v, v2, v3,v4} =
{c, e} {b, c,g,d} = {e}. Since we have
A(v5) {Vi, V2, V3, V4, V5} = {d} {b,c,g,d,e} = 0,
A(v4)- {v1,v2,v3, v4, v5} = {c,e}-{ b,c,g,d,e} =0,
A(v3) {vi, v2, V3, v4, v5} = {c} {b,c,g,d,e} = 0,
A(v2) {v1,v2,v3,v4,v5} = {b,g,d} {b,c,g,d,e} = 0, and
A(vi) {v,,v2, v3, v4, v5} = {a,c,f} {b,c,g,d,e} = {a,f },
v6 must be either a or f. Let v6 = f. Then we have that
A(v6) {vi,v2,v3,v4,V5,V6} = {b}- {b,c,g,d,e,f} =0,
A(v5)-{v1,v2, v3,v4, v5,v6} = {d}- {b,c,g,d,e,f} = 0,
A(v4) {vi,v2,v3,v4,v5,v6} = {c,e} -{b,c,g,d,e,f} =0,
A(v3) {v1,v2,v3,v4,v5,v6} = {c} {b,c,g,d,e,f} = 0,
A(v2)-{ fV1,v2, v3,v4, v5, v6} = {b,gd} {b,c,g,d,e,f} = 0, and
A(vi)- {v-1,v2, v3, v4,v5, v6} = {a,c,f}-{b,c,g,d,e,f} = {a}.
Therefore v7 must be a; that is, we construct a vertex sequence
(V1, V2,...,V7) = (b,c,g,d,e, f,a).
For a given tree we will get different vertex sequences if we choose different
vertices whenever we have two or more choices. But the sequence t = (b, c, d, e, f, a, g)
cannot be a vertex sequence of T above. In fact, we have v5 = f. But we have that
A(v4) {vi,v2, v3, v4} = {d} {b,c,d,e} = 0,
A(v3) {Vi, V2, V3, V4} = {c, e} {b, c, d, e} =0,
A(v2) {Vi,V2, V3, V4} = {b,d,g} {b,c,d,e} = {g}, and
A(vi)-{V1,V2,V3,V4} = {a,c,f}-{ b,c,d,e} = {a,f}.
That is, the sequence t has f, one of the elements of A(vi) {v1, v2, v3, v4}, for v5
instead of g which is an element of A(v2) {v1, v2, v3, v4}. It is a contradiction to
the maximality of k in the definition of vertex sequence.
Proposition 3.3.7 Let (v1, V2,..., V,) be a vertex sequence of an n-tree T. For any
vj with j > 1 there is unique 1 < k < j such that vj E A(vk) {v1,v2,... vj-}.
Proof First of all, for any j > 1, we have vj 5 vi for i = 1,2,... ,j- 1 from the
Definition 3.3.4. Assume that there is j > 1 such that


{vj, vj+I,...,vn} n ((Ujl-'A(vi)) u {Vl, v2,..., vj1-}) = 0.








Then, for any u E {vj,,vj+i,.. .,vn} and any w E (U=I-A(v,)) U {v1,v2, ..,vj},
there is no (u,w)-path in T. That is, T is not connected. It is a contradiction
to the connectedness of a tree. Therefore there is 1 < k < j such that vj E
A(vk) {Vi,V2,...,vj-i}. If there are 1 < ki < k2 < j such that vj E A(vk,) -
{v1i,v2, ...,vj-l} and vj E A(vk2) {Vi, V2,.. .,vj-i}, then the two edges, {vj, vk}
and {vj, vk2}, are elements of E(T). There is a path between vk, and vk2 in the
subtree {{vl, v2,... vj-}, E'} where E' = {{u,v} E E(T)j u,v v {V1, V2, .. -,Vj-}}.
The tree T contains a cycle consisting of {vj, Vk,}, {vj, Vk2} and the (Vkl, Vk2)-path.
This contradicts the fact that a tree is acyclic. Therefore there is only one 1 < k < j
such that vj E A(vk) {vi,V2,... ,vj_}. I
From the Definition 3.3.4 and the previous lemma we get the following corol-
lary immediately.
Corollary 3.3.8 Let (v, v2,... v,n) be a vertex sequence of an n-tree. Then
(1) v2 E A(vi)
(2) If d(vj) > 2 for any j = 2,3,..., n 1, then Vj+l E A(vj).
(3) For any j = 2,3,... ,n- 1 if d(vj) = 1 and there is i < j such that d(vi) >_ 3
and d(vk) = 2 for every k = i + 1, i + 2,... ,j 1, then vj+l E A(vi) and Vj+l .
A(vp), for every p= 1,2,...,i- 1,i + 1,...,j.
Proposition 3.3.9 Let (Vi, v2,... Vn) be a vertex sequence of an n-tree and A(vi) =
{vli,vI,12...,vli} with 1 <11 < 12 <...< lm. Then
{V, V2, Vn} = UhlB(vi, vi,).
Proposition 3.3.10 Let = (1, v2,..., v,.n) be a vertex sequence of a given n-tree
T. If d(vj) = m for some j > 1, then there exist m vertices, vi, vj,,vj2,...,viji
such that A(vj) = {vi, vj,,vj2,...,vimi} with i Proof For any j > 1 there exists unique i < j such that {vi,vj} E E(T)
by Proposition 3.3.7; that is, vi e A(vj). So, for any vk A(vj) {vi}, j < k.







Since d(vj) = m, there are m 1 vertices, Vj Vj2... ,vj_,, such that A(vj) =
{vi, Vjl, Vj2 ,..., vj.I} with < j < j < j2< ... < jm-. J-

Proposition 3.3.11 Let = (V1, v2,..., v,,} be a vertex sequence of a given n-tree
T. Let d(vi) = m and A(vi) = {V11,V12,. ,Vi} be the adjacent set of v1, named
so that 1 <11 < 12< ... < 1m. Then, for any i = 1,2,... ,m, the set of vertices in
the subbranch, B'(vi,vii), forms a section in s starting with vii,.

Proof Clearly, the section containing vi, starts with v1, = v2. Let B'(vi,v11) =
{VI,V11,+i,. .. ,Vl+1m}. Assume that there is a vertex u E V(T) B'(vi,vi,) such
that (v11,v11+1-,...,v11+j,u,...) is a sectionof 5. Since u V B'(vi,vi,), A(vll+k)-
B'(vi,vl,) = 0 for any k = 0,1,2,... ,j from the definition. Thus U=0A(vl1+k) -
B'(vi, vl) = 0. That is, Uk=oA(vl,+k) D B'(vi,vi,). On theother hand, A(vll+k) C
B'(vi,vl,) forever k = 0,1,2,... ,j. Thus wehavethat U=o=A(vl+k) C B'(vi,vi).
Therefore U =A(vi+k) = B'(vi,vl,). That is, j = mi. Hence B'(vl,vl) forms

a section in s. Since B'(vi,vi) is a subbranch of vI, Vli+mi+l V A(u) for any
u E B'(vi,vl). Thus vll+mi+l E A(vi). That is, Vll+mi+1 = v12. For any i =
1,2,... ,m 1, let B'(vi, vi,) form a section in s and vl,+m+i = v1i+1. By a simi-
lar method, B'(vi,vli+,) forms a section in s and Vi;+,i+m +l1 = Vii+2. I

The section formed by a subbranch B'(vi,vi) is denoted by (B'(vi,vl,)).
From the previous proposition, we have the following corollaries.
Corollary 3.3.12 Let A = (v1, V2, ... Vn) be a vertex sequence of a given n-tree
T. Let d(vi) = mn and A(vi) = {vi,v12,...,vin} be the adjacent set of vi, named
so that 1 < 11 < 12 < ... < 1m. The section (B'(vi,vi,)) precedes the section
(B'(vi,vi,+,)) in A for i = 1,2,...,m 1, and there is no other vertex between
these two sections.
Corollary 3.3.13 Let (vi,V2,... V) be a vertex sequence of a given n-tree T.
For any j > 1 with d(vj) = m > 2 let A(vj) = {vi,v,,v12,..., ,,_i} where i <








j < Ji < J2 < ... < jm-1. Then (B'(vj,vj,)) = Vi,vji+ ...,vj(i,+l)-1) for any
i = 1,2,..., m 1. In particular, we have that ji = j + 1.

Note that B(vj, vjm ,) = {vj, Vjm_, Vj.,i+i,..., Vj.-i } is the (vj, vj,,_ )-branch
vertex set for some jm > jm-1.

Corollary 3.3.14 Let (vl,v2, ... vn) be a vertex sequence of a given n-tree. If
i < j with vj A(vi) and vj 3 A(vk) for every k = i + 1,i + 2,...,j- 1, then
vp A(vk) for every p > j and k = i+ 1,i + 2,...,j- 1.

For any given vertex sequence s = (Vi, v2,..., vn) and any vertex vj 7$ vi, Vi
belongs to a branch of vj. We want to give a special name to this branch.

Definition 3.3.15 Let = (vi, V2,- vn) be a vertex sequence of a given n-tree
T. For any j > 1 the root branch of vj in is the branch of vj containing vU.

Note that there is no root branch for vj.

Proposition 3.3.16 Let (vi, v2,. .. v,) E S be the vertex sequence set of a given
n-tree T. For any 1 < j < n, if vj+i A(vj), then vj is an end vertex.
Proof If vj+l V A(vj) for some 1 < j < n, then vj has a unique edge with a
vertex, vi, with i < j by Proposition 3.3.7. That is, d(vj) = 1. Therefore vj is an
end vertex. L

Proposition 3.3.17 Let (vi,v2,. ,vn) S be a vertex sequence set of a given
n-tree T. For any 1 < j < n there is an integer jm such that
{Vl,U2, ...,vj} U {Vjm,vjm+i,...,vn}
is the vertex set of the root branch of vj.
Proof Let 1 < j < n and d(vj) = m. Then, by Proposition 3.3.10, there exist
m different vertices, vi,vj,,vj2,... ,vjm_ such that A(vj) = {vi,vjl,vj2,.... vii}
with i < < < j (u=l B(vj, vj,)) = (,vjl ,..., ).








That is, the union of all the subbranches of vj except the root branch is a section
from vj, to Vj-i.- Therefore the set of vertices in the root branch is {v1, v2,... vj}U
{VJm,Vj.m+l,-- ..,vn}. IL
Note that, if jm > n, then the root branch is {V1, v,..., vj}.

Corollary 3.3.18 Let = (V1, v2,. vn) E S be the vertex sequence set of a given
n-tree T. Let j 54 1. Then d(vj) = 1, that is, vj is an end vertex of T, if and only
if the set of vertices of the root branch of vj is V(T) = {v1, v2, ... Vn}-

Proposition 3.3.19 Let = (v1, v2,... ,vn) be a vertex sequence of a given n-
tree T. For any vertex vj with j > 1 the (vi,vj)-path in T is a subsequence of

(Vi, V2, .. vj) .
Proof For any vertex vj with j > 1 there is a unique vertex Vkp with kp < j
such that vj E A(vkp). For the vertex vkp there is another unique vertex vkp_,
with kp-1 < kp such that Vkp C A(vkpi). Then we will have that v, is the unique
adjacent vertex of a previous vertex Vk,. Therefore the (vi, vj)-path is a subsequence
(vi,vkI,...,vk_,vkfp,vj) of (vl,v2,...,Vj). -L

Proposition 3.3.20 For any two vertex sequences S, and s2 in the v-vertex se-
quence set S, = {s E SIV1 = v} the root branch vertex sets of a vertex x :A v, = v
in s1 and in s2 are same.

Proof Let S1 = (vi, v2, ,v,n) and g2 = (ui,u2,..., Un) be two different vertex
sequences of a given tree T with v, = ul. Let x be an arbitrary vertex in T
with x =A v1. Then there are two integers k and I such that Vk = x = ul. Let
A(x) = {xi,x2,... ,xm-1,y}. Since T is a tree, there is a unique (vi,x)-path in T.
Then one of the elements of A(x) is in the (vi, x)-path. Without loss of generality,
let y be in the (vi, x)-path. From the proof of Proposition 3.3.17 we have that
(U j'=B'(x,x,)) = (u 'B'(vk,xp))
= (Vk+l, Vk+2,. ,Vk+r} and
(U=1 B'(x,xp)) = (Um='B'(ul,xp))
= (U+i,UL+2,. ,UI+r)








for some r. Therefore we have that
{Vl,V 2,...,Vk} U {Vk+r+l,..., Vn} = {u1,U2,..., U} U {Iu+r+l,..., n}.
That is, the root branch vertex sets of x in g, and in 2 are same. _

Example 3.3.21 Let T be an 8-tree as shown below.

v w


p q r s t U
Figure 3-3

Clearly, s = (q,r,s,t,u,w,v,p), S2 = (q,p,r,s,w,t,u,v) 5r = {f E
Siv1 = q}. Let
(vi,v2,...,v8) = ).
Then the root branch vertex set of s is {p,q,r,v}. That is, {p, q,r, v} = {q,r} U

{v,p} = {vi,v2} U {v7,v8} and {p,q,r,v} = {q,p,r} U {v} = {u1,u2,u3} U {us}.
Therefore the root branch vertex sets are same but the position might be different.
On the other hand, d(w) = 1 and the unique adjacent vertex of w = V6
is s = v3. Thus s = V3 is in the root branch of w = v6. Therefore the root
branch vertex set of w = v6 is {vi,v2,v3,v4,V5, v6} U {v, v8} = {vi, v2,..., V8s} =
{q,r,s,t,u,w,v,p}. That is, the root branch vertex set of an end vertex is V(T)
itself.

3.4 Degree Sequence

In this section, we define the degree sequence of a vertex sequence.

Definition 3.4.1 Let S be the vertex sequence set of an n-tree T. For any ver-
tex sequence = (V1,V2,. .,V n) E S, we call Di = (dl,d2,.. .,dn) the degree se-
quence of where d, = d(vi) and di = d(vi) 1 for i > 2. We call the set
D = {Djl. E S} the degree sequence set of S. And, for any vertex v E V, we








call DV = {DjsA E S,} C D the v-degree sequence set of S,. For any section
(B(vj, vj,)) = (vj, vj,,vj,,+,...,vj,+_i) of A, formed by the (vj,vj,)-branch vertex
set B(vj,vj,), we define Bd(vj,vj,) = {dj,djdj,+i,...,dj,+1_i} as the (vj,vj,)-
branch degree setwhere dk = d(vk) 1 for k = j, ji, j +1, .. ,ji+ 1. And we define
the (vj, vj.)-branch degree section by (Bd(vj, vj,)) = (dj, dj,, dj,+1, dj,+ I) which
is denoted by (Bd(vj,)) if there is no confusion. In the same way as in Definition
3.3.1, we define (Bd'(vj,)) = (Bd'(vj,vj,)) = (djdj,+...,dj+i-1)

Example 3.4.2 Occasionally, it may happen that there are two or more vertex se-
quences of a given tree with the same degree sequence. In the following example, there
are several pairs of vertex sequences with the same degree sequences, respectively.

q


p r .5
Figure 3-4

Since there are 5 vertices in V(T), there are exactly 5! different sequences of vertices
of T having length 5 and using each vertex only once. But only 16 of them satisfy
the condition (*) in Definition 3.3.4. Therefore the vertex sequence set, S, of the
given tree, T, consists of the following sequences
i = (p,r,q,s,t), s = (r,p,q,s,t), 813 = (s,r,p,q,t) ,
2 = (p,r,s,t,q), s8 = (r,p,s,t,q), s4 = (s,r,q,p,t),
S3= (q,r,p,s,t), 9 = (r,q,p,s,t), s15 = (s,t,r,p,q),
S4 = (q,r,s,t,p), Sio= (r,q,s,t,p), s6 = (s,t,r,q,p),
i5 = (t,s,r,p,q), S3i = (r,s,t,p,q),
46 = (t,s,r,q,p), s32 = (r,s,t,q,p).
Then the degree sequences of these vertex sequences are
D-, = (1,2,0,1,0), D-, = (3,0,0,1,0), D.i3 = (2,2,0,0,0),
Dj- = (1,2,1,0,0), D-, = (3,0,1,0,0), Dsj, = (2,2,0,0,0),
D-3 =(1,2,0,1,0), D-9 =(3,0,0,1,0), Dsj5 =(2,0,2,0,0),
= (1,2,1,0,0), D,,0 = (3,0,1,0,0), D,6 = (2,0,2,0,0),
S=(1,1,2,0,0), Dh =(3,1,0,0,0),
D =(1,1,2,0,0), D,2 =(3,1,0,0,0).








Now we have that

Dj- = D-3, D,- = Di4, D5, = D4, D,-, = Ds,
Di, = D8l0, Dqs = D82, Dsi3 = Di4, Ds15 = Dai6.

That is, for some different vertex sequences, the degree sequences might be same.

And, if the degree sequences are same, then the trees denoted by these degree se-

quences are isomorphic.

Remark: The sum of the terms in any degree sequence of a given n-tree is the
same, namely, n 1. It is 4 in the above example, and it is same as the number of
edges in the tree. See Theorem 3.4.6 below. Since d, = d(vi) and d, = d(vi) 1 for

i = 2, 3,..., n, we have the following proposition.

Proposition 3.4.3 Let Di = (d,d2,. ,dn) be a degree sequence of a vertex se-

quence s of an n-tree T. Then 0 < di < n-1, and 0 < di < n-1 for i= 2,..., n.

Proposition 3.4.4 If dj 5 0 for any 1 < j < n, then vj+l is an adjacent vertex
of vj.

Proposition 3.4.5 Let = (v1,v2,..., Vn) be a vertex sequence of an n-tree T and
Ds = (di,d2,. ..,dn) the degree sequence of A. For any j E {1,2,...,n 1} with
dj = m > 1, if the set {vj+1, Vj+2,..., Vk} is the collection of all the vj-subbranches

except the root subbranch, then k > j + m.

Proof Since dj = m = d(vj) 1, from Proposition 3.3.10, the vertex vj has m

subbranches besides the root subbranch. Thus there are at least mrn endvertices in
the collection of those subbranches other than the root branch. Therefore at least m

terms in (dj,dj+1,...,dk) are 0. Hence k > j + m. .

Theorem 3.4.6 (First Reduced Degree Sum Formula) For any degree sequence
Di = (dl,d2,..., dn) of a vertex sequence A E S of a given n-tree T we have

Z1 di = n l = \E
and


d + E-2(d 1)= 0.








Proof d, is the degree of a vertex, and the remaining n 1 terms are each one
less than the degree of a vertex. By the degree sum formula (Proposition 3.2.7), it is
n
true that Ei=1 d(vi) = 2\E\. So
nn
j=1 di = d + Ei=2 di
= d(vi) i Z,=2 (d(vi) 1)
^+ En En
= d(vi) + Ei=2 d(v) Ei=2 1
=(d(vi) + =2 d(v)) -(- 1)
= d(vi)- (n -)
= 21E (n 1)
=2(n 1) (n 1)
=n- 1= E|.
Then we have that
d, + En2(d 1) d, + En2 E=21
iE=(-l =d+ i=2d i=21
=Eni=1 di- (n 1)
=(n-i)-(n-i)
0.-

Corollary 3.4.7 For any degree sequence, Dj = (d1d2, ...,dn) of a vertex sequence
A E S of a given n-tree T, we have
d + ,=2, dO(di 1) = E 2, d,=0 1.

Proof From the first reduced degree sum formula we have that d1+Ei=2(d-1) = 0.
Then
0 = d + E=2(d 1)
= d1 + Ei=2, do(di 1) + E=2, d,=0(d 1)
= d + Ei=2, do(di 1) + Ei=2, d,=O(-1)
= d + Ei=2, d,0(di 1) Ei=2, d=o 1.
Thus we get the result. I

Note that di > 0 for any i = 1,2,...,n. So di 1 < 0 if and only if di = 0.

Corollary 3.4.7 shows that, in a degree sequence D (di, d2,..., dn), the total
number of 0 terms is the same as di + S"-=2 (di 1). Therefore there are exactly
di:O
di + En=2 d,0(di 1) endvertices in the given tree. Because of this, we have the
following proposition.

Proposition 3.4.8 For any degree sequence Ds = (dl,d2,...,dn) of a vertex se-
quence A E S of a given n-tree T, we have








dj + Ez=2(d 1) # o
for every 1 < q < n.
Proof Assume that there is q < n such that di + EZ 2(d 1) = 0. Since D =
(dl,d2, ...,dn) is a degree sequence of a vertex sequence s E S of the given n-tree
T, we have that vq+1 A(vp) for some p < q. Thus at least one of the endvertices
counted by Vp is one of Vq+l,Vq+2,... ,Vn. That is, in the section (d1, d2,. dq),
q (i- 1), that is,
we have at least one less 0 terms than di + Ei=2, dO(di ) a is
d + q (i_1) -_ =2 1 q1
d + Ei=2, dO(dj i) 2 d,=O -
Then
0 =d+E?=2(d- 1)
d q+Z- (di )+Z2 =O(d? 1)
= d + EL:2, d0i ( d 1) + E?2, d,=o(- 1)
= d + EL2, d4o(di -1) + E d=o(-)
i=2, di i=2, d,=O('
>1.
This is a contradiction. Therefore we have di + E-2(di 1) 5 0 for every 1 < q <
n. -
Theorem 3.4.9 (Second Reduced Degree Sum Formula) Let T = (V,E)
be an n-tree. Let Ds = (d1,d2,...,dn) be the degree sequence of a vertex sequence
A E S of the given n-tree T. For any jk > 1, let B'(vj,) = {Vj,Vjk+I,...,Vjk+1-1}
be a (vj, vjk)-subbranch vertex set. Then we have
dj + -(di 1)=0
dk+..i=jk+l -
and
djk + EZ=+l(di 1) 5 0

for every jk < q < jk+I.
Proof Let E(B'(vjk)) = {{u,v} E E(T)ju, v E B'(v,k)}. Then
T(vj,) = (B'(vjk),E(B'(vjk))) c T
is also a tree, and (B'(vjk)) = (VjkVjk+l,... ,Vj+,l_) is a vertex sequence of the
tree T(vjk). Note that d(vjk) = dj +1 in T. Since vj B'(vjk), the edge {V, Vjk}
is not in the edge set E(B'(vjk)). Thus we have that d(vjk) = dj, in the subtree








T(vJk). Then the result comes from the first reduced degree sum formula (Theorem
3.4.6) and Proposition 3.4.8. 1l

Corollary 3.4.10 Let A = (V1,V2,...,v,,) be a vertex sequence of an n-tree T and
Ds = (d,d2,. ,d,d) the degree sequence of s. For any dj > 0, if the subsequence
(vj, v+l,... vp) is the collection of all the vj-branches except the root subbranch of
vj, then
dj + E +(d 1)= 0
and
d3 +E=j,+l(di 1) 0
for every j < q < p.

Proof For j = 1, they are true by the first reduced degree sum formula (Theorem
3.4.6) and Proposition 3.4.8. Let j > 1 and vj E A(vt) for some t (i) If dj = 0, then d(vj) = 1, that is, vj is an end vertex. Since j > 1, vj 5 vi.
Thus the root branch is the whole tree by Corollary 3.3.18. Therefore (vj) is the
collection of all the vj-branches except the root subbranch and dj + EZ-=j+ (d 1) =
dj = 0.
(ii) If dj > 0, then the subsequence (vj, vj+l,..., vp) which is the collection of all
the vj-branches except the root subbranch, is the (vt, vj)-subbranch, since vj E A(vt).
By the previous theorem, we get dj + E-=j+l (di 1) = 0. Thus the collection of all
the vj-branches except the root subbranch of vj is the (vt, vj)-subbranch. Therefore,
by the second reduced degree sum formula, we get dj + Eq+( 1) 0 or every
+~ +1 d 1) jf 0 for every

j < q < p.

Not all of the nonnegative integer sequences with n entries can be a vertex
sequence of an n-tree. There is a condition to being a degree sequence of an n-tree.

Definition 3.4.11 Let D = (dl,d2,..,d, ) be a nonnegative integer sequence.
We say that D satisfies the degree sequence condition, briefly DSC, if and only if it
satisfies all of the following:








(a) di > 0.
(b) E', di = n 1.
(c) For any j < n with dj = m > 0, there are exactly m + 1 integers, j < ji =
J+l djk + Ejk+''(di 1) = 0 for k = 1,2,...,m.
(d) For any jk with dj = I > 0, there are exactly + 1 integers, jk < JkA =
ik +1< k2 < ... < Jk, < Jk(,+1) = Jk+1 such that
d ,P+ Ej+)l(d -1) 0 for p = 1,2,..., 1.

Theorem 3.4.12 Let D = (,dl ,..., dn) be a nonnegative integer sequence. Then
D is a degree sequence of a vertex sequence of an n-tree if and only if D satisfies
DSC.

Proof [=] This follows from Proposition 3.4.3, Theorem 3.4.6 and Theorem 3.4.9.
[=] Let V = {v1,v2,... ,v,} be a set of n different vertices. Let D =
(di, d2,.. .,dn) be a nonnegative integer sequence satisfying the above conditions.
Since d, > 0 and di > 0 for any i = 2,3,..., n, we can assign the degree of each
vertex vi E V by d(vi) = d, and d(vu) = di + 1 for i = 2,3,...,n. Let j < n and
dj = m > 0. From (c), there are exactly m + 1 integers, j < j1 < J2 < ... < jm <
jm+i, such that ji = j + 1, and dj, + EZij+(di 1) = 0 for k = 1,2,...,m. Now
we assign an edge between vj and vj, for k = 1,2,...,m. That is, vjk E A(vj)
for i = 1,2, ...,m.
(i) If djk = > 0, we assign an edge between vjk and vjk for p = 1,2,...,1
by (d). We do a similar procedure for each vertex, vj, assigned to dj > 0. Therefore
vj, C A(vj) and vjP EA(vj,) for k= 1,2,...,m, j= 1,2,...,l.
(ii) If dj, = 0, then we do not assign any new edge for the vertex vi,. But it is
already joined to the vertex vj by an edge.
Let s = (vi, v2,..., vn). Then all the vertices in s are connected. Since each vertex
vi is assigned di new edges and >F>' di = n 1 from (b), we have n vertices and








n 1 edges. Then the set of all the edges assigned above, together with the vertex

set V, forms an n-tree by Proposition 3.2.6. 1

Remark: The proof of Theorem 3.4.12 shows how to construct a tree that corre-

spond to the given degree sequence satisfying DSC.

Example 3.4.13 Let D = (3,2,3,1,0,0,0,1,0,2,0,0, 0) be a nonnegative integer

sequence having 13 entries. Let V = {V1, v2,..., vn,} be a set of n different vertices.


V9


Y12 V13 V8 V7



VlO V1 V2 V3 V4 V5


Vl1 V6
Figure 3-5

(a) Clearly, Ei31 di = 12 and d, = 3 > 0. So we expect a 13-tree.

(b) Since d(vi) = 3, there are 4 different integers, 1 < 11 = 2 < 12 = 10 < 13 =

13 < 14 = 14, such that d2+Ei=3(j-1) = 2+(2+0-1-1-1+0-1) =

0, d0io + E =11(di 1) = 2 + (-1 1) = 0, and d13 = 0. We assign edges between

v, and each of v2, V10, v13.

(c) (i) Since v2 = 2, there 3 different integers, 2 < 3 < 8 < 10, such that d3 +

Si=4(d i-1) = 3+(0-1 1 -1)=0, and d8+(d9-1) = 1-1 =0. Weassign
edges between v2 and each of v3 and vs.

(ii) Since v0io = 2, there are 3 different integers, 10 < 11 < 12 < 13, such that

dll = 0 and d12 = 0. We assign edges between vl0 and each of vll and v12.

(iii) Since v13 = 0, we do not assign any new edge for v13. But the vertex v13

is connected to vi by an edge.








In a similar way, we get {v3, v4}, {v3, v6}, {v3, V7}, {Vs, vg}, and {v4, v5}. Let E be
the collection of all the edges above, that is,
E = {{vi, V2}, {Vi, V0io}, {Vi, V13}, {V2, V3}, {vJ2, V8}, {Vio0, V1},

{v10, V12}, {v3, v4}, {v3, v6}, {v3, v7}, {v8, v9}, {v4, v5}}.
Since the edge set E contains 12 elements, the set T = {V, E} is a 13-tree.

3.5 Perfect Sequence and Main Theorem

In the previous section, we showed that, for a given n-tree T, there exist many
vertex sequences depending on the starting vertex and choice among the adjacent
vertices for each successive vertex. Therefore the degree sequence set D for the
given tree T contains many different degree sequences which denote the same tree
T. We need a way to choose one degree sequence representing the tree T. First of
all we want to define an order on the set of all finite, nonnegative, integer sequences.

Definition 3.5.1 For any two finite nonnegative integer sequences D = (Pl, P2, ..., IPk)
and E = (ql, q2,..., qi), D < E if and only if either there is m < min{k, l} such
that pm i = 1,2,...,k. We write D < E if D < E or D = E.

Proposition 3.5.2 Let D be the set of all degree sequences of an n-tree. The degree
sequence set, (D, <), is a totally ordered finite set.

Proof For any two different degree sequences the order defined above is well defined.
Since each term di of a degree sequence is nonnegative and less than n and the
length of each degree sequence is n, the degree sequence set D is a finite set. Thus
(D, <) is a totally ordered finite set. I

Definition 3.5.3 For a given n-tree T, let S be the vertex sequence set of T and
D the degree sequence set of T. Then the maximum degree sequence in D is called
the perfect sequence of the given tree T, and it is denoted by pf(T). For the perfect
sequence, pf(T), of a given tree, T, there is a vertex sequence s = (V1i, v2, ..., Vn) E








S such that the degree sequence of A is pf(T). We call A a perfect vertex sequence
of the tree and v, a perfect vertex of the tree.

In the Example 3.4.2, the perfect sequence is (3,1,0,0, 0) and there are two
different perfect vertex sequences, s\ = (r, s, t, p, q) and 412 = (r,s,t,q,p) for it.
But the trees represented by snl and s12 are isomorphic.

For any given tree the perfect sequence is unique by Proposition 3.5.2. But
there may exist several different perfect vertex sequences. If the given tree is a
labeled tree, then these perfect vertex sequences denote different labeled trees. But
these labeled trees are isomorphic to each other; that is, there is a unique unlabeled
tree. Therefore, for a given perfect sequence, there exists a unique unlabeled tree.

Theorem 3.5.4 Let dj be the perfect sequence of a tree T. If T' is any other tree
whose perfect sequence is also dj, then T' is isomorphic to T.

Proof If T has more than one vertex sequence with the same degree sequence, then
these vertex sequences all produce the isomorphic trees, by using the construction in

the proof of Theorem 3.4.12. 1
Because of this theorem, we have our main theorem below.

Theorem 3.5.5 (Tree Classification Theorem) For any positive integer n, let

T(n) be the set of unlabeled n-trees and P(n) the set of perfect sequences of n-trees.
Then there is a one-to-one correspondence between T(n) and P(n).

3.6 Obtaining Perfect Sequences

By Theorem 3.5.5, if we can construct the set of perfect sequences of n-trees,
P(n), without being given any tree shape, then we can classify all the n-trees by this
set. And, in the same way as in Example 3.4.13, we can get the different tree shapes
from the elements of P(n).

Proposition 3.6.1 For the perfect vertex vl of a perfect vertex sequence A =

(vlV2,...,vn), we have d(vi) > d(vi) for i = 2,3,...,n.








Proof If there is some k {2,3,...,n} such that d(vk) > d(vi), then D- =
(d(vi),...) < D, = (d(vk),...) for any vertex sequence = (Vk,...). This contra-
dicts the maximality of Dj. _

Proposition 3.6.2 If D = (dj, d2, ..., d) is the perfect sequence of a given n-tree
T, then n -1 > dl > di > dn = O forany i=2,3,...,n-1.

Proof Since T is an n-tree and Ds is the perfect sequence, we have n- 1 > dl. For
any vertex u E V(T) we have d(vi) > d(u) by Proposition 3.6.1. Since di = d(vi)
and di = d(vi) 1 for every 1 < i < n, we have d, > cdi for 1 < i must denote one of the end vertices, each of which has degree 1, we have dn = 0.
Therefore we have di > dn = 0 for every 1 < i < n- 1. l

Definition 3.6.3 Let D = (di, d2,..., d) be a degree sequence. Then we say that
D satisfies the ordered subbranch condition, denoted OSbC, if and only if D satisfies
the following condition for all j> 1 :
If d(vj) = m + 1 > 2, and B'(vj,), B'(vj2), ..., B'(vj) with j < ji < j2 <
S m< JM are the subbranches of the vertex vj other than the root subbranch, then
(Bd'(vjk,,)) > (Bd'(vjk+l)) for every k = 1,2,...,m- 1.
Recall that (Bd'(vj,)) is a section of the degree sequence D. (See Definition
3.3.3 and 3.4.1.)

Remark : If j = 1, then there is no root branch of vi; instead, there is one more
subbranch, Bd'(Vjm+,), such that jm < jm+1 and (B'(vjm)) > (Bd'(vj,,)).

Theorem 3.6.4 Let D = (dj, d2,... dn) be a nonnegative integer sequence. If D
is the perfect sequence of an n-tree, then it satisfies the DSC and the OSbC.

Proof It is clear from Theorem 3.4.12 and the maximality of D. I

Proposition 3.6.5 Let T be a tree and v a vertex with the largest degree in T.
Then there is exactly one degree sequence D, for T starting at v and satisfying
both DSC and the OSbC.









By the above proposition, the converse of Theorem 3.6.4 is not true. The

following example is a counterexample.

Example 3.6.6 Let A = (dj,d2,...,d12) = (3,2,2,1,0,0,2,0,0,1,0,0). Then

Z'i di = 11 and d, = 3 > 0. And the sequence A can be separated into 4
sections like (3), (2,2,1,0,0,2,0,0), (1,0), (0). The second section can be sep-

arated into 3 sections, (2), (2,1,0,0), (2,0,0), and the second and third sec-

tion of these can be separated by (2), (1,0), (0) and (2), (0), (0), respec-

tively. All of these subsections satisfy the second and third conditions of DSC.

Thus the sequence A satisfies the DSC. That is, the sequence A is a degree se-

quence of a vertex sequence (Vi, v2, ... v12) of a 12-tree by Theorem 3.4.12. On the

other hand, the sequence A satisfies the OSbC, since we have (2,2,1,0,0,2,0,0) >

(1,0) > (0). But A is not the perfect sequence of the 12-tree. In fact, we rearrange

the sequence A to get B = (dr + 1,d2, d -1, dio, dn,d12,d3, d4, d5,d6, ds, d9) =

(3,2,2,1,0,0,2,1,0,0,0,0). Note that d(vi) = d, and d(v7) = d7 + 1. Clearly the

sequence B also satisfies the DSC and the OSbC. So B is also a degree sequence

of another vertex sequence (V7, v2, v1, V10, v11, V12, v3, v4, v5, v6, V8, vg) of the 12-tree.

That is, A and B are degree sequences of the same 12-tree (see Figure 3-6). And

we have

B= (3,2,2,1,0,0,2, 1,0,0,0,0) > (3,2,2, 1,0,0,2,0,0,1,0,0) = A.

Thus A is not the perfect sequence. Furthermore A cannot be the perfect sequence

of any tree, because A and B denote isomorphic trees, and B > A.


V11 V10 V1 V12
V9

V2
V7
V8
V5 V4 V3 V6
Figure 3-6








In this example we have that d(vi) = 3 = d(v7) which is the largest degree
in the tree. That is, the perfect sequence must start with the largest degree. We will
see in Example 3.6.8 that the degree sequence B is the perfect sequence of this tree.
Note that (1, 0) is the only possible degree sequence when d, = 1. Therefore
we assume d, > 1 from now on.
Now we show how to construct alternate degree sequences from A, similar
to B. Assume that we have a nonnegative integer sequence A = (di,d2,... ,d.)
satisfying the DSC and the OSbC. If there is 1 < i < n such that d, < di, then
A cannot be the perfect sequence of any tree by Proposition 3.6.2. So we assume
that d, > di for i = 2,3,... ,n, and there is at least one integer 1 < j < n such
that dj = cd -1. That is, there is an n-tree T = (V, E) such that A is a degree
sequence of a vertex sequence A = (V1, V2,... V,v) of T, and T has at least two
vertices of maximum degree dl. Clearly, dj > 0. Thus vj is not an end vertex in
T. By Corollary 3.3.18, the set of the vertices of the root branch of vj is not the
set of vertices in A itself. So, from Proposition 3.3.17 there is j < m < n such
that RB(vj) = {Vi,V2,...,vj} U {Vm+i,Vm+2,. ..,Vn} is the set of the vertices of
the root branch of vj. Clearly, (Vj+i,Vj+2,... ,Vm) is the section containing all of
the subbranches of vj except the root branch. Therefore, if dj = p, then there are
p subsections of A like (Bd'(vj,)) > (Bd'(vj,)) > ... > (Bd'(vj,)) with j < ji <

j.2 <... < jp < m+1, since A satisfies OSbC. On the other hand, let E(RB'(vj)) =
{{u,v} E E(T)ju,v E RB'(vj)} be the set of edges in the root subbranch of vj. Then
T' = {RB'(vj),E(RB'(vj))} is a subtree of T. Let S(T') be the vertex sequence
set of T'. By Proposition 3.3.7, there is a unique integer 1 < k < j such that
Vk E A(v,). Clearly, vk E RB'(vj). Let Svk(T') be the vk-vertex sequence set of
S(T'), in which the first term of each vertex sequence is vk. Let Dvk(T') be the
vk-degree sequence set of Svk(T') (see Definition 3.4.1). Since Dv,(T') is a finite
set, there is a maximum sequence in Dk (T'). We denote the maximum sequence in








Dvk (T') by mdR(vk). Note that d, must be changed to d, 1 in mdR(vk), since
it is not going to be used for the first term anymore in the new degree sequence.
Now we have that either mdR(vk) > (Bd'(vj,)), or there is an integer 1 < q < p
such that (Bd'(vjq)) > mdR(vk) > (Bd'(vjq,+)). Note that mdR(vk) may be the
smallest section, then vjq1 would not exist.
We rearrange the given degree sequence A to a new sequence in which dj + 1
is the first term and the subbranch degree sequences of vj,
mdR(vk), (Bd'(vj,)) (Bd'(vj2)) ..., (Bd'(vj)),
are listed in descending order. Denote this new degree sequence by MA.
Since the subbranch degree sequences of vj are listed in descending order in
MA the new degree sequence MA satisfies the OSbC. Since MA is a rearrangement

of A with di =d(vj) =dj+l1 >0, d, = d(vi)- 1 = di -1 for some i> 1, and
A satisfies the DSC, the new degree sequence MA satisfies the conditions (a) and
(b) of the definition of the DSC. Since mdR(vk) is the perfect sequence of T', it
satisfies the conditions (c) and (d) of the definition of the DSC. Clearly, each section,
(Bd'(vj,)), satisfies the conditions (c) and (d) of the definition of the DSC. Therefore
the new degree sequence MA satisfies the DSC. Thus the new degree sequence MA
V) Vi
satisfies both the DSC and the OSbC.
For the given nonnegative integer sequence A = (dj, d2, ... ,dn), the set
{M\Id(vj) = d(vi)} is a non-empty finite set, since A E {MAId(vj) = d(vl)}.
Therefore there is a maximum element in this set. Let PA = max{ MA d(vj) = d(vi)}.
Now A determines a unique tree T. Suppose B is another vertex sequence for T
satisfying the DSC and the OSbC. Then {.1'Id(vq) = d(vi)} = {M d(v) = d(vI),
where Vq is an analogue of vj, by Proposition 3.6.5. Therefore PA is the perfect
sequence for T.

By the above explanation together with Proposition 3.6.2 and Theorem 3.4.12,
we have the following theorem.








Theorem 3.6.7 Let A = (di, d2,. ., d,) be a nonnegative integer sequence satisfy-
ing the DSC and the OSbC with di > d, for i = 2,3,...,n. Then A = PA if and
only if A is the perfect sequence of an n-tree.

In the following example, we show the degree sequence B in Example 3.6.6
is the perfect sequence of the tree.

Example 3.6.8 In Example 3.6.6,
A= (dl,d2,...,d12) =(3,2,2,1,0,0,2,0,0,1,0,0)
satisfies the DSC and the OSbC and d2 = d3 = d7 = di 1. That is, d(v2) = d(v3) =
d(v7) = d(vi). Thus max{MAIj = 2,3, 7} is the perfect sequence of the tree. For
v7 we have that RB(v7) = {v1, v2, .. ,v7} U {vo10, V11, V12} and (Bd'(vs)) = (0) =
(Bd'(v9g)). Thus T' = {RB'(v7),E(RB'(vr)} is the subtree produced by the root
subbranch of Vy. (See Figure 3-7.)
Since v7 C A(v2), we get the vr-vertex sequence set

Sv,(T') = { (v2,Vi,v1OV11,V12,v3,v4,v5, v6)
(v2, Vl, v10, v11, v12, v3, v6, v4, v5)
(v2, Vi, v12, Vlo, 11, v3, v4, V5s, v6)
(v2, Vl, V12, Vlo, vii, V3, v6, v4, V5)
(v2, v3, v4, V5, v6, V1, Vio, vii, v12)
(v2, V3, V4, v5, v6, V1, V12, v10, vii)
(v2, v3, V6, v4, v5, V1, vio, Vll, v12),
(V2, v3, v6, V4, V5, Vl, V12, vo, V11) }.
Then the v7-degree sequence set of S,,(T') is

D,7(T') ={ (2,2,1,0,0,2,1,0,0),
(2,2,1,0,0,2,0,1,0),
(2,2,0,1,0,2,1,0,0),
(2,2,0,1,0,2,0,1,0) ,
(2,2,1,0,0,2,1,0,0),
(2,2,1,0,0,2,0,1,0),
(2,2,0,1,0,2,1,0,0),
(2,2,0,1,0,2,0,1,0) }
= { (2,2,1,0,0,2,1,0,0),
(2,2,1,0,0,2,0,1,0),
(2,2,0,1,0,2,1,0,0),
(2,2,0,1,0,2,0,1,0) }.








Therefore we have the maximum sequence mdR(v7) in D, (T') as mdR(v7) =

(2,2,1,0,0,2,1,0,0). Since mdR(vr) = (2,2,1,0,0,2,1,0,0) > (0) = (Bd'(vs)) =

(Bd'(v9)), we have MA = (3,2,2,1,0,0,2,1,0,0,0,0).

In the same way, we get

MA = (3,2,1,0,0,2,1,0,0,2,0,0)
MV3 =(3,2,2,1,0,0,2,0,0,1,0,0).

Therefore max{Mt,,j = 2,3,7} = M17. Since

MA =(3,2,2,1,0,0,2,1,0,0,0,0)
> (3,2,2,1,0,0,2,00,0,1,0,0)
= A,

M4, which is B in the Example 3.6.6, is the perfect sequence of the tree.


VlI VIO V1 V12


V2


V5 V4 V3 V6
Figure 3-7


By the Tree Classification Theorem (Theorem 3.5.5) and the previous example,

we have the following important corollary.

Corollary 3.6.9 Let A = (d, d2,. .., dn) be a nonnegative integer sequence satis-

fying the DSC and the OSbC. If d, 1 > di for every i = 2,3,..., n, then A is the

perfect sequence for an n-tree.

We will see in the following example how we construct a tree shape from a
given maximal degree sequence.

Example 3.6.10 Let D = (di, d2,..., d0io) = (3,2,1,2,0,0,1,0,0,0) be a maximal

degree sequence. Then we expect a 10-tree, T. Let s = (V1, v2,... vio) be the vertex

sequence having D as the degree sequence. Then d(vi) = d, = 3, so the vertex v,









has 3 edges. (See Figure 3-8.) Though there is no specific order for these edges, we

label them as eC, e2, e3 for the convenience of explanation.



e2

e3 V1 el
Figure 3-8


From the definition of vertex sequence (Definition 3.3.4) v2 is one of the

elements of the adjacent set of v1, A(vl). Without loss of generality, let el =

{vI,v2}. Then v2 has 2 'new' edges, e4 and e5, since d2 = 2. Let e4 = {v2,v3}.

By the same reasoning, v3 has 1 'new' edge, named e6. So e6 = {v3, v4}. Since

d4 = 2, V4 has 2 'new' edges, named e7 and e8. (See Figure 3-9.)



e2 e5 e8

e3 V1 el V2 e4 V3 e6 v4 e7
Figure 3-9


Since d5 = 0 and d6 = 0, neither v5 nor v6 has a 'new' edge; that is, V5

and V6 are end vertices. On the other hand, v5 and v6 are the adjacent vertices of

v4, since d4 = 2. Thus V7 is an adjacent vertex of one of {vi,v2,v3}. Since d3 = 1

and v4 is the adjacent vertex of v3, v3 is not an adjacent vertex of v7. Since

d2= 2 and we have only one adjacent vertex of v2, namely 1V3, we add an adjacent

vertex, v7. (See Figure 3-10.)


V7 V6

C2 C5 e8

e3 v, e1 V2 e4 v3 e6 V4 e7 V5
Figure 3-10








Since d7 = 1, v7 has 1 'new' edge which connects with v8. Since d8 = 0,
there is no 'new' edge from V8. Since we covered up to d2, the remaining two vertices,

v9 and v0io, are adjacent vertices of v, which has two 'unused' edges, e2 and e3.

Let e2 = {v1, v9}, and e3 = {vi, vIo}. Since we are thinking about an unlabeled tree
and we put the edge names for our convenience, there are no specific names for the

vertices and edges. Thus we get the following 10-tree. (See Figure 3-11.)


(V8)


(V9) ()(V6)



(Vio) (V1) (V2) (V3) (V4) (V5)
Figure 3-11


3.7 Extended Partitions and the Algorithm

In Definition 3.5.3 and the following explanation, it was shown that any tree

can be represented by a unique perfect sequence. So, in this section, we develop an

algorithm to produce all the perfect sequences for the n-trees for any positive integer

n. From these perfect sequences, we get the shape of each n-tree as shown in the
previous example, as well as the total number of n-trees.

From Definition 3.4.1 and the First Reduced Degree Sum Formula (Theorem

3.4.6), we see that a degree sequence for an (n + 1)-tree always has n + 1 nonnegative
integers as entries and these integers always add up to n which is the number of

the edges of the given (n + 1)-tree. Thus, our first step in listing all possible degree
sequences is to form ordered partitions of the integer n as sums of smaller positive
integers.








Definition 3.7.1 An ordered partition of the positive integer n is a sequence of
positive integers, (pi,P2,... ,Pm) such that E=1 pi = n.

Note that 1 < m < n in the definition above.

Definition 3.7.2 An extended partition of an ordered partition (p1,p2,. ,Pm) of
the positive integer n is a sequence of n + 1 nonnegative integers, (di, d2,..., ,n+)
such that
(i) di = pi, E+1 di = n, and
(ii) there is a subsequence (dj, dj2,..., dim) such that djk = Pk for k = 1,...,m.

Note that any entry di in the extended partition must be 0 unless it is a
member of the ordered partition. It is important that different extended partitions
are produced not only by the different ordered partitions but can also be produced
by the same ordered partition.
For example, let (di,d2,... dn+,i) and (di,d,.... ,dn+i) be two extended
partitions of an ordered partition (pi,p2,. ..,Pm), and let (djl, dj2,... ,dj,,) and

dl, d'2,... ,d'm) be subsequences, respectively, such that djl = d = =i = d\t and
djk =Pk = dt, for k = 2,3,... ,m. Assume that j, $ ti for at least one i. Then
there are sections of zeros with different lengths between d(k-l) and dj, and d'
and d' respectively. So the extended partitions are different. Thus the order is
very important in an extended partition.
Also note that the length of the extended partition of n must be n + 1. So,
for any given ordered partition, (p, P2,... ,Pm), we need to add n m + 1 zeros to
make an extended partition.

Proposition 3.7.3 Let (dj, d2,... ,dn+,) be an extended partition of an ordered
partition P = (djl,dj2,... ,djm) of the integer n. Then this extended partition sat-
isfies the First Reduced Degree Sum Formula (Theorem 3.4.6).








Proof From the given extended partition (di, d2,... dn+i) of an ordered partition
Ei"+l di dilJ, =n and
(dj dj2,..., d.,) of the integer n, we have that Z 1i = = n and
i + d (- n+1
~ i+ l2(d -1) d + Ek=2(dik -1) + ELi=, di.P(di -1) since d1 5 0
d= d+ Ek=2(dJ- 1) + E 1, dp(d, 1)
+ n+1
= EiM_ d3k + E + E+ 2+=1, d p(-1)
= n + (m 1)(-l) + (n m + 1)(-l),
=n-m+l-n+m- 1=0.I

Even though we have a very large number of extended partitions of n, some of
them can not be a maximal degree sequence for an (n + 1)-tree. So we first eliminate
some ordered partitions that could not possibly result in maximal degree sequences.
To do so, we need to check some properties of perfect sequences.

Proposition 3.7.4 Let Ds = (d1,d2,... ,dn+) be the perfect sequence of a given
(n + 1)-tree T. If di, 0 and di+1 = 0, then 0 = di+1 = di+2 = ... = di+d,.

Proof Since di, 0, the vertex vi has di 'new' edges; that is, there are di adjacent
vertices of vi and they are labeled by d, consecutive integers after i. Since Di is
a perfect sequence, it satisfies the OSbC. That is, the vertex vi+1 is a vertex of the
largest degree among the adjacent vertices of vi. But di+1 = 0. Therefore we have
dj =0 for j = i + 2, i + 3,..., i + di. .

This proposition gives us a method to make an extended partition from an
ordered partition by adding zeros. That is, we know certain extended partitions
cannot be perfect sequences just by checking this property.

Proposition 3.7.5 Let Di = (dl, d2,. ., dn+) be the perfect sequence of a given
(n + 1)-tree T. If n > 2, then d > 2.

Proof If d, = 1, then di = 0 for every i = 2,3,... ,n + 1 by Proposition 3.6.2.
Since n > 2, there are at least 3 vertices in T. Thus we have that di+1-.=(di-1) =
1_+21(-1) < 1+(-1-1) = -1 0. This is a contradiction to the First Reduced
Degree Sum Formula. Therefore d, > 2. 1_








According to the above proposition, the only trees whose perfect sequences

start with 0 or 1 are the tree with zero edge and the tree with one edge. The only

degree sequence representation of a tree with zero edge is (0), and the only degree

sequence representation of a tree with one edge is (1,0). These are easy enough to

visualize and count, so for the algorithm we work on the assumption that we are

listing trees with at least two edges. Then the first term of the perfect sequence will

be at least 2.

The following proposition shows that we have an arc shape tree if the first

term of the perfect sequence is 2.

Proposition 3.7.6 If Di = (dia, d2,..., dn+) is the perfect sequence of an (n+ l1)-tree

T and d, =2, then di = 1 for i = 2,3,...,n 1 and d4 = dn+, =0.

Proof By Proposition 3.6.2 we have that di, E {0, 1} for any i = 2,3,... ,n + 1,

since di = 2. Then the possible highest degree in T is degree 2. So T is just a

linear tree with n edges. That is, there are exactly two end vertices which cannot be

v1. Therefore there are exactly two zeros in the sequence. Write a degree sequence

for T, in the order of the vertices in the following figure.


--- --- --- ---
Vn+l Vi V2 V3 Vn-1 Vn
Figure 3-12


That is, start at a vertex adjacent to an end vertex and move in the longer

direction first. Clearly the degree sequence resulting from this method contains

the two zeros for vn and Vn+l. Then we will get a degree sequence of the form

D = (2, 1,1,..., 1,0,0). For the same tree, the numbers in a degree sequence cannot

change, but only the order in which they occur can change by changing the position

of zeros. And, clearly, the new ordered sequence < D. So the sequence D must be

the perfect sequence of T. That is, Di = D. I








Most of the extended partitions of n start with a 1 or a 2. Since the last two

propositions classify trees whose degree sequences start with these values, we do not

need to consider ordered partitions that start with 1 or 2.

Proposition 3.7.6 shows that for any given number n of edges there is only one
tree with n edges whose perfect sequence starts with a 2. In particular, it is a linear

tree with n edges, since d, = 1 for i = 2, 3,..., n 1. We add this sequence at the

end of the algorithm.
By Proposition 3.6.2, we do not need the ordered partitions in which the largest

summand appears more than once or is not the first term in the ordered partition.

If an ordered partition has either one or two entries only, then we can produce

only one extended partition for each case. See the following Proposition 3.7.7 and

Corollary 3.7.8.

Proposition 3.7.7 Let D- = (dl, d2,..., dn+) be the perfect sequence of an (n+l)-

tree T with d2 = 0. Then d, =n and d, = 0 for all i > 1.

Proof: Let A be a vertex sequence yielding a perfect sequence Ds of an (n + 1)-

tree T. Since Ds is a perfect sequence and 0 = d2 = d(v2) 1, all of the vertices in

the tree other than vi have degree 1, so all the terms in D, other than d, must be

zero. By the First Reduced Degree Sum Formula (Theorem 3.4.6), we have d, = n. 1_

Corollary 3.7.8 If an ordered partition P = (pl,p2) of n has only two entries,
then the produced extended partition, (di, d2,... ,dn+1), from P is such that d, =

p1, d2 =p2 and di = 0 for i = 3,4,...,n + 1.

We demonstrated several necessary conditions which apply to a perfect se-

quence of a tree. Unfortunately, while these conditions are sufficient in almost all
cases, there are certain sequences which, although they satisfy all the conditions, are

not perfect sequences of the trees they represent. The problems usually arise when
the tree has two or more vertices of the largest degree. We saw a counterexample in

Example 3.6.6.








Note that in the Example 3.6.6 the two degree sequences A and B have the

same ordered partition, P = (3,2,2,1,2,1). That is, our algorithm produces the

degree sequence B as one of the extended partitions of the ordered partition P.

Therefore all we need to do is just drop the degree sequence A from the result of

our algorithm.

Each step described so far has involved checking each member of a list of

possible degree sequences and removing those which could not be the perfect sequence

of a tree. The last step is no exception. Given a degree sequence, we check if there is

more than one vertex of maximal degree. If there are more than one (in Example 3.6.6,

both vertices v, and v7 = u1 have the same degree, 3), we write degree sequences

for the tree starting at each such vertex. If the largest of these with respect to

the ordering < is the sequence we are checking, then it is a perfect sequence. If

some other vertex gives a larger degree sequence, it is false that the sequence we are

checking is a perfect sequence. This is a surefire approach in that, given a degree

sequence, it checks for a perfect sequence by considering every possible alternative

for the maximum and seeing if the sequence in hand is the largest.














CHAPTER 4
TREES AS INVARIANTS FOR BRAIN TUMOR SHAPES

4.1 Introduction

A sphere packing treatment plan for a given brain tumor is produced in the

first chapter. This plan can be represented numerically as a list of parameters of

the dose spheres. In particular, the list consists of the coordinates of the dose center

in millimeters along the lateral (Lat), anterior-posterior (AP), and axial (Ax) axes

together with the radius of the dose sphere in millimeters. From these data, we

can assign a unique corresponding graph by matching a sphere with a vertex and

matching the adjacency of two spheres with an edge. Then, by using the notion of

cutvertex, we give an order to the vertex set. We use this order to decide which edges

to choose in order to obtain a unique maximal tree contained in the graph.

Our main theorem of this section is Theorem 4.5.6: The perfect sequences are

invariants of the shapes of arbitrary brain tumors. We then develop a Mathematica

program (see Appendix B) to get a maximal tree of the graph for a sphere packing

plan for a given brain tumor. In the near future, we plan to use this program to

classify all the treatment plans which have been done at the Brain Institute of the

University of Florida. We hope that the classification gives some insight about brain

tumor shapes.

In this chapter, we assume that a brain tumor is connected, so the graph

representation for any brain tumor is a connected graph.








4.2 Definitions and Some Properties

Definition 4.2.1 A graph G = (V(G),E(G)) consists of a vertex set V(G) =
{Vi,.. .,vn} and an edge set E(G) = {ei,...,em} C {{u,v}lu,v E V(G)} in which
each edge is an unordered pair of two distinct vertices u and v, and the edge {u, v}
is said to join u and v. For an edge {u, v} we say that u and v are adjacent
vertices and the vertex u and the edge {u, v} are incident with each other, as are
v and {u,v}. From now on, IV(G)I denotes the number of vertices and jE(G)j
denotes the number of edges. For a vertex u V(G), the vertex set
AG(u) = {v E Gjv is an adjacent vertex of u}
is called the adjacent set of u in G.

Since there is no loop represented by a single edge, in a graph in this article,
no vertex is adjacent to itself.

Definition 4.2.2 We say that a graph G is isomorphic to a graph H if and only
if there exists a bijection f: V(G) -* V(H) such that {u, v} C E(G) if and only if
{f(u),f(v)} I (H).

Definition 4.2.3 A walk is an alternating sequence (vo, el, vi, e2,..., ek, Vk) of ver-
tices and edges s.t. vi-1 J vi for i = 1,2,... ,k, and ei = {vi-i,vi} for every i. A
cycle is a walk (vo, el,v1, e2,..., ek, Vk) with k > 3 in which v0 = vk is the only
vertex repeated. A path is a walk with no repeated vertex. A (u,v)-path is a path
with the first vertex u and the last vertex v. A (u,u)-path is a vertex u. The length
of the (u,v)-path, l(u, v), is the number of edges in the (u,v)-path. So the length of
(u,u)-path is 0. The distance d(u,v) between two vertices u and v is the length of
a shortest path joining them, if any; otherwise d(u, v) = oo.

Note that a path of length > 1 is of linear shape.

Definition 4.2.4 Let G = (V, E) be a graph where V = {vi,v2,...,v"n}. Then,
for any vertex v E V, we say that PG(v) = j if and only if








j = min{i {1,2,..., n}I there is a (v, vi)-path in G}.
We call Pc(v) the path-value, or simply the P-value of the vertex v in G.

Definition 4.2.5 Let G = (V, E) be a graph. For any nonempth subset V' C V,
let F = { {u,v} E E I u,v E V } C E. For any subset F' of F, the graph
G' = (V', F') is called a subgraph of G. A graph G is connected if and only if every

pair of vertices are joined by a path. A graph having no cycle is acyclic. Note that an

acyclic graph can be disconnected. A forest is an acyclic graph. A tree is a connected

acyclic graph. An n-tree is a tree with n vertices.

Definition 4.2.6 The degree of a vertex v, denoted by d(v), is the sum of the

number of edges incident with v.

Definition 4.2.7 A cutvertex of a graph is a vertex of degree greater than 2, whose

removal increases the number of components of the graph.

Definition 4.2.8 A nonseparable graph is connected, nontrivial, and has no cutver-
tex. A block of a graph is a maximal nonseparable subgraph.

Thus if v is a cutvertex of a connected graph G, then G- v is disconnected.

Note that the traditional definition of cutvertex (usually called cutpoint in graph

theory) does not have the degree condition. But we do not want to separate any

linear shape subgraph into several blocks because there is no difficulty in handling
linear shape subgraphs in our algorithm. Thus we do not call a vertex with degree

2, a cutvertex in this chapter, even though it separates the graph.
And if H is a block, then none of the vertices in H is a cutvertex of H.

Note that if there is no cutvertex in G, then the graph G itself is a block. This
definition differs from the definition of block in Harary [5], because we have a different
definition of cutvertex. Because of this, we have Proposition 4.2.10 below.

Proposition 4.2.9 If a graph G is of linear shape, then the maximal tree of G is

G itself.








Proposition 4.2.10 [5] Let v be a vertex of a connected graph G. The following
statements are equivalent:
(1) v separates G and d(v) > 2.
(2) There exist vertices u and w distinct from v such that v is on every (u,w)-
path.
(3) There exists a partition of the set of vertices V(G) {v} into subsets U and W
such that for any vertices u E U and w W, the vertex v is on every (u, w)-path.

Definition 4.2.11 Let G = {V, E} be a graph. For any vertex v E V we define
the edge subset IG(v) C E(G) by IaG(v) = {{v,u} E E(G)I u E AG(v)}. We call
IG(v) the incident edge set of v in G.

Clearly, Ia(v) contains all the incident edges of the vertex v in G.

Definition 4.2.12 Let G = {V, E} be a graph. For any subset F of E we call the
set V(F) = {v E V(G)I there is an edge e E F such that v and e are incident}
the F vertex set.

Proposition 4.2.13 [5] The following statements are equivalent for a graph T:
(1) T is a tree.
(2) Every two vertices of T are joined by a unique path.
(3) T is connected and IV(T)I = IE(T)I + 1.
(4) T is acyclic and JV(T)J = JE(T)| + 1.

Note that if a tree has n vertices, then there are exactly n 1 edges.

Definition 4.2.14 Let G = {V, E} be a connected graph and E' a subset of E.
A connected subgraph T = {V, E'} is called a maximal tree of G if and only if
T'= {V, E'U {e}} contains a cycle for any edge e E E E'.

Our work in this chapter produces a particular maximal tree for any given
graph. So we have the following (well-known) theorem.

Theorem 4.2.15 Every connected graph has a maximal tree.





63

Example 4.2.16 Let G be a graph with 14 vertices and 18 edges as in the figure
4-1 below.


a b c d e f


Figure 4-1


There are 3 cutvertices, b, c, e. So the given graph G can be separated into
6 blocks as follows:
G1 ={Vi,Ei}={{a,b }, {{a,b}}},
G2 = {V2, E2} = { { b,c, k, ,m,n },
{ {b,c}, {c, k}, {k, {, m}, {mn}, {rn, b}, {b, k}, {k, m} } },
G3 ={V3,E3}={{c,d,e}, {{c,d},{d,e}}},
G4 = {V4, E4} = { { e,f,g }, { {e,f},{f,g},{g,e} } },
G5 = {V5,E5} = { { e,h,i }, { {e,h},{h,i} } }, and
G6 ={V6, E6}={{e,j }, {{e,j}}}.


a b


b


c d e
G3


9


e f
G4


Figure 4-2








4.3 Graphs for Sphere Packing Plans

The sphere packing treatment plan S having n spheres for a given brain tu-

mor, from Chapter 1, consists of the centers, c1, c2,..., cn, and the radii, ri, r2,..., r.,

of the spheres with ci = (xi, yi, zi) for every i, where x, is for the AP coordinate,

y, is for the Lat coordinate and zi is for the Ax coordinate of the center, ci. (Note
that the physicians use diameters, rather than radii, in their data sets.)

We consider the spheres to be the vertices of a graph, and we assign the vertex,
vi, for the sphere centered at ci with radius ri. Now we can assign an edge between
two vertices if the spheres assigned by these two vertices meet. In Chapter 1, we

noted that the collimators do not deliver clear cut boundaries for spheres-there is

some spillover. The neurosurgeons currently assume that the portion between two

spheres gets enough dosage of radiation if the distance between the centers of the

two spheres is less than or equal to eleven tenths of the sum of the radii of the two
spheres. Thus for any two vertices assigned for two spheres satisfying this condition
we assign an edge between them. Then we have the following proposition.

Proposition 4.3.1 Let G = {V,E} be a graph for the sphere packing plan of a

brain tumor. For any vertices vi and vj in V, d(ci,cj) < (ri + r) if and only if

{vi, vj} E E. Thus we have the following edge set;
E = {{i,j}Id(ci, cj) < (ll/10)(ri + rj) where i < j, i = 1,2,...,n 1}.
Since any brain tumor is connected in this article, there is at least one (u, v)-

path in G for every two vertices, u and v in G. Thus we have the following propo-

sition.

Proposition 4.3.2 The graph G = {V, E} produced by the above method for a
given tumor is a connected graph.

Since we are dealing with unlabeled graphs in this article, we have the following
proposition.








Proposition 4.3.3 For any sphere packing treatment plan for a given brain tumor,

there exists a unique graph that we assign to the treatment plan.

By means of the above method, we get a unique connected graph G = {V, E}

for the sphere packing treatment plan for the tumor. This graph does not have the

information about the positions of the dose spheres as an aspect of the structure and

looks only at how the spheres are attached. It is clear that the graph G does not

have a loop or multiple edges between two vertices, since every edge is assigned for

two different vertices which are assigned for two attached spheres.

4.4 Order in the Vertex Set

To get a maximal tree from a graph we may need to delete some edges in the

graph. Thus we need to label each vertex in order to choose certain edges to delete,

even though we are dealing with an unlabeled graph in this chapter. So there is no

specific meaning for this labeling except that it is only used to choose a maximal tree

for the graph.

Example 4.4.1 Let P be the sphere packing treatment plan (see Figure 4-3) for a

given brain tumor.







Figure 4-3


Then, by Proposition 4.3.3, there is a unique graph G for the sphere packing plan.

The graph is given below in Figure 4-4.


p -- s



Figure 4-4








There are two different isomorphism classes, namely T1 and T2, of maximal trees

for the graph G. That is, we could choose either of these for a maximal tree to assign

the graph G. Since, in our algorithm, we want to get the same maximal tree for a

given graph every time, we need certain rules to get the same tree. (See Figure 4-5.)






T, T
Figure 4-5


In this chapter, the graph comes from a sphere packing treatment plan. That is,

the main shape of the brain tumor depends on the size of spheres and the connection

between spheres. Even though we cannot keep the information about the size of

spheres in the graph, it seems to us that the bigger the sphere is, the more it has a

chance to get attached to more spheres. So we assume that the larger degree vertices

play a more important role in classifying the tumor shapes into trees than the smaller

degree vertices. That is, by choosing a largest degree vertex first, the shape of the

tumor is apparently most closely preserved. Thus, in the previous example, we choose

the vertex q or s as the starting vertex; then we pick all the incident edges. Then

we get the tree T1. Therefore we want to choose the tree T1 for the maximal tree of

G. Note that the tree T2 produces a linear graph which does not show the shape of

the tumor as closely as T1.

But there are some vertices which are more important than the bigger degree
vertices. Recall the graph G in Example 4.2.16. Then, in the block G2, it is clear

that the vertex k has the largest degree, 5. But the cutvertices, b and c, play a

critical role in obtaining a maximal tree of G. So, to get a maximal tree for a graph

G, we want to start at the cutvertices of G first, instead of the vertices with the

largest degree. So we want to label the cutvertices first. On the other hand, there






67


are different kinds of 'cutvertices' in some graphs. The following example shows such

a case.

Example 4.4.2 Let G = {V, E} be the graph in Figure 4-6.


U3


Figure 4-6


Then there are two cutvertices of the graph G, namely u2 and uj0. Us-

ing these two cutvertices, the graph G is separated into four subgraphs, namely

G1, G2, G3 and G4. (See Figure 4-7.)


1 G2
"II*------10


U3 G3



U1
U2 U2


U9


U10 UIO/ \ U


U2
Figure 4-7


Then the subgraphs G1, G2, and G3 are blocks, but the subgraph G4 con-

tains its own cutvertices, u4 and u8, which are not cutvertices of the graph G. We

separate the subgraph G4 into four subgraphs by using its cutvertices u4 and us.








(See Figure 4-8.) Note that there is no specific order in labeling the subgraphs. So

at this moment, we relabel the subgraphs of G by H1, H2, ..., H7.

U9
H2 U10 U10 / \




UU0 U8
u3 //H 5
H1 / /

H4
U2 U2 U2 U4
U8 U5 ___ u
U7


H6 u6 H7


-- -U5
U4 U4
Figure 4-8

Then, again, the subgraphs H1, H2, ..., H6 are blocks, but the subgraph H7

contains its own cutvertex, u6, which is neither a cutvertex of the graph G nor a

cutvertex of the subgraph G4. We separate this subgraph H7 into two subgraphs by

using U6. (See Figure 4-9.) And we relabel the subgraphs of G by B1, B2, ..., B8.

Therefore {u2, u10} is the set of cutvertices of the graph G. But {u4, us} is

the set of the cutvertices of a subgraph which is produced after the first separation

using the cutvertices of the graph, and u6 is the cutvertex of a subgraph which is

produced after the second separation.

For the purpose of labeling the vertices in a graph, we separate these cutvertex

sets for different levels, from each other.

Definition 4.4.3 Let G be the graph assigned for a given sphere packing plan

P. Then the vertex set V(G) of the graph G is the union of two disjoint subsets,

namely CI(G) and Cc(G), where Ci(G) = {v V(G)I v is a cutvertex of G}






69


and Cc(G) = V(G) CI(G). We call C,(G) the first step cutvertex set of G. For

every i = 2,3,..., IV(G)I, we define a subset Ci(G) of CcI(G) as the collection

of the vertices of Cc I(G), which are cutvertices of a subgraph produced after the

(i 1)-th separation using the elements of Ci-i(G). Then we call C1(G) the i-th

step cutvertex set of G, and let C.(G) = Cic(G) Ci(G). We assume that there

are finitely many vertices in a given graph. If there exists at least one cutvertex,

then there exists an integer 1 < k < n such that Ci(G) 54 0 for every i < k, and

Ci(G) = 0 for every i > k + 1. Then we call k the separation step constant of the

graph G, and let Cc(G) = (Ck)c(G).


If there is no cutvertex of G

separation step constant of G is 0.


B2 u10

U11-


U3



U1
U2


U8



B6


then the graph is a block, and we say that the


tU4
Figure 4-9.


U9


U10 U8
B5


B4
U2 U4

U7




U6


-- U5


Proposition 4.4.4 Let JV(G)j = n and let the separation step constant of G be

k > 0. Then there is a sequence of integers {ml,m2,...,mk} with 0 < mn1 < m2 <

... < rnk such that 1CI(G)I = mi, and \Ci(G)j = mi, min1, for i = 2,3,...,k.


^,







Proof Since k > 0, the first step cutvertex set Ci(G) is not empty. Thus there is
a positive integer, namely mi, such that ICi(G)| = mi. For every i = 2,3,...,k,
let mi = mi-1 + |C1(G)|. For every i = 2,3,... k, the i-th step cutvertex set Ci(G)
is not empty. So we have that mi > mi-1, and ICi(G)| = mi mi-,. I

Now we label the vertices as follows;
(1) the vertices in Ci(G) are labeled by {1,2,...,mI},
(2) for every i = 2,3,..., k, the vertices in Ci(G) are labeled by {mi-i + 1, mi-i +
2,...,mi 1,mi}, and
(3) the vertices in CC(G) are labeled by {mk + 1, mk + 2,..., n}.
We arbitrarily name the vertices of Ci(G) and CC(G), and denote that CI(G)
= {Ui,U2,...,Uml}, Ci(G) = {Um,_i+I,Um,_i+2,...,Um,} for i = 2,3,...,k, and
CC(G) = {Umk+1,Umk+2, ... ,Un}. It is clear that for every vertex up E Ci(G) and
every vertex Uq E Cj(G) with i $ j, we have that p < q if and only if i < j.
And also, for every vertex up E Ci(G) for every i = 1,2,... k, and every vertex
Uq E Cc(G), we have that p < q. Assume that, for every vertex ui, the center and
the radius of the sphere which is assigned to that vertex ui, are ci = {xi, yi, zi} and
ri, respectively.
We now relabel the elements for each of the sets, C1(G), C2(G),..., Ck(G)
and CC(G), in the following way:
for every pair of vertices, ui, uj, in one of CI(G),C2(G),... ,Ck(G) and Cc(G),
i < j if and only if
(i) d(ui) > d(uj); or
(ii) If d(ui) = d(uj), then r, > rj; or
(ii) If d(ui) = d(uj) and r, = rj, then xi < xj; or
(iii) If d(ui) = d(uj), ri = rj, and x, = xj, then yi < yj; or
(iv) If d(ui) = d(uj), r = rj, x, = xj, and y = yj, then zi Note that d(ui) represents degree of the vertex u, in G.







It is clear that for any two different spheres, at least one of the coordinates is
different. Thus any two distinct spheres can always be compared by the above order.
Therefore the above order for the vertices of the graph is well defined.

4.5 Maximal Tree from a Graph

In this section, we develop an algorithm that will uniquely select a maximal
tree from the graph which is produced for a given brain tumor. The choice is made
so that the tree will most closely resemble, in our opinion, the shape of the tumor -
that is, of the graph obtained for the tumor.
Let G = (V, E) be the graph for a given sphere packing plan. Let V(G) =
{u1, u2,..., uI. for some positive integer n, where all the vertices in V are labeled
by the method in the previous section.
Define a subgraph To = (V(G),Fo) with V(To) = V = V(G) and F0 =
0; that is, there is no edge in To. Then, for every i = 1, 2,... ,n, we have that
PTo(ui) = i. (See Definition 4.2.4.)
(1) For the vertex ul, let F1 = F0 U IG(ul). (See Definition 4.2.11.) Let T1 =
(V(G), F1). The subgraph (V(F1), F1) C T1 is a tree. Thus, if V(F1) = V(G), then
T, = (V(G), F1) = (V(F1), F1) is our maximal tree of G. Note that for every vertex
v E V(F1) we have that PTi (v) = 1, and for every vertex u, V V(F1), we have that
PT, (Uj) = J.
(2) Assume that V(F1) 5 V(G). For the vertex u2, there are two cases, that is,
PTi (u2) = 1 and PT1(U2) = 2. Now define the edge subset I'(u2) by I(u2) =
IG(U2) {{u2,v} V E\ PT1(v) = 1)}.
(a) If PT (u2) = 1, that is, there is a (ui,u2)-path in the subgraph T1, then let
F2 = FiUIG(u2). Each edge in these {{u2, v} El PT, (v) 1}= {{u2, v} E Ely E
V(F1)} makes a cycle with {u1,u2} and {ui, V}, since PT, (U2) = 1. Thus there is
no cycle in the subgraph (V(F2),F2), so it is a tree. Let T2 = (V(G),F2) C G.







If V(F2) = V(G), then T2 = (V(G),F2) = (V(F2),F2) is our maximal tree of G.
Note that for every vertex v E V(F2) we have that PT2(V) = 1, and for every vertex
uj 3 V(F2) we have that PT2(Uj)= j.
(b) If PTi(u2) = 2, then there is no (Ul,U2)-path in the subgraph T1.
(i) If there is a non-empty subset {u21,, U22,... U2} C V(IG(u2)) where PT1(U2,)
=1 for i = 1,2,...,t and 2i <2j for l {{u2,u21}}. Then the subgraph (V(F2),F2) C G is a tree. Let T2 = (V(G),F2) C
G. If V(F2) = V(G), then T2 = (V(G),F2) = (V(F2),F2) is our maximal tree of
G. Note that for every vertex v E V(F2) we have that PT2(v) = 1, and for every
vertex uj g V(F2) we have that PT2(Uj) = j.
(ii) If none of the vertices in V(IG(u2)) has p-value 1, that is, for every vertex
v E V(IaG(u2)) there is no (ul,v)-path, then let F2 = F1 U IG(u2). Since PT,(w) = 1
for every vertex w E F1 and PT, (v) = 2 for every vertex v E IG(U2), the subgraph
(V(F2), F2) is a forest, but not a tree. Let T2 = (V(G),F2) C G. It is clear that
the subgraph T2 cannot be our maximal tree, even if we have that V(F2) = V(G),
because the subgraph T2 is disconnected. Note that for every vertex v E V(F2) =
V(F1 U IG(U2)) = V(F1) U V(Io(u2)) we have that PT2(v) = 1 if v E V(F1), and
PT2(v) = 2 if v E V(IG(u2)). And, for every vertex uj V(F2), we have that
PT2(UJ) j.
(3) By induction, at the j-th step, the subgraph Tj_j = (V(G),Fji) has been
defined. We want to define the subgraph Tj C G. For the vertex uj, there are two
cases, that is, PT,_-(uj) = j and PT,_I(uj) = q for some 1 < q < j. For every
i = 1,2,..., n, define
mj_(i) = min{l {1,2,... ,n}|PT_,(ul) = i and ul E AG(u) }.
If the set {l E {1,2,...,n}juj E V(IG(uj)) and PT,_(ui) = i } is empty, then the

vertex u,_,,_(i) is not defined. So there is no edge {uj,Um,_(i)}.







(a) Let PT,- (uj) = ; that is, for every i = 1,2,... ,j- 1,j + 1,...,n, there is
no (uj,,ui)-path in the subgraph Tj-i = (V(G), Fji). Let
Fj = Fj-, U (U={ {uj,um ,(i)} })
Then the subgraph (V(F,),Fj) is a forest. Let Tj = (V(G), F,) C G. If V(F,) =
V(G) and IV(Fj)l = iFj| + 1, then, by Proposition 4.2.13, the subgraph Tj =
(V(G), F,) = (V(F,), F,) is our maximal tree of G.
(b) Let PT,_-(uj) = q for some 1 < q < j, that is, there is a (uq,Uj)-path in the
subgraph Tj,- = (V(G), Fj-i). Now let
Fj = Fj-1 U (U'1, ,iq{ {U,,Uj _,(i)}})
Then the subgraph (V(Fj), F) is a forest. Let T, = (V(G),Fj) C G. If V(Fj) =
V(G) and IV(Fj)t = IFj| + 1, then by Proposition 4.2.13, the subgraph Tj =
(V(G), Fj) = (V(F,), Fj) is our maximal tree of G.

Note that if u is an i-th step cutvertex, then d(u) in G is same as d(u) in
Ti. Also note that, for every i = 1,2,...,n, {uiu2,...,ui} C V(Fi), that is,
V(G) = V(Fn). We will see in Theorem 4.5.3 that Tn = (V(G), Fn) = (V(Fn),F")
is our maximal tree of G. By the Tree Classification Theorem (Theorem 3.6.8), this
tree Tn can be denoted by a unique perfect sequence.
Example 4.5.1 Let G = (V,E) be a graph where V = {u,u2,.. .,u8}, which
are ordered as in the previous subsection, and E = {eli e2, ,..., e13} as in the figure
below. Clearly, the graph G does not have a cutvertex, so the order is decided by
only the sizes of the spheres and the coordinates of the centers of the spheres.

us e13 U?7 e2 "U1


U2 C5 U6
Figure 4-10









Let To = (V(G),Fo) be a subgraph of G = (V,E) where Fo = 0, as shown in

Figure 4-11.


U8 U7 U1
0 U

U4
U3. **K


*
Z2 Z6
Figure 4-11

Since Ia(ul) = {e1,e2}, let F = FoUIG(ul) = {e1,e2}. Then Y(F) = {u1,u5,u7}-

Since V(Fi) = {u,u5,u7} 5 V(G), the tree (V(F1),F1) is not a maximal tree of

G. Let T1 = (V(G), F1) which is shown in Figure 4-12.


Us U7 e2 U1

e6l
U30 U4
UU5



Z2 U6
Figure 4-12

We have that mi(i) = i for i = 1,2,3,4,6, and 8, but mi(5) and m1(7) do not

exist. Since PT,(U2)= 2 and Aa(u2) = {u3,u4, u6}, let

F2 = F U (U,=3,4,6{{u2,Ui}})
= F1 U IG(U2)
= {e, e2} U {e3, e4, e5} = {fe1,e2, e3,e4,e5}.

Then V(F2) = {ui,u5,U7,U2,uu 4,U6U} $ V(G) and the subgraph (V(F2),F2) is a

forest, not a tree. Thus it cannot be a maximal tree of G. Let T2 = (V(G), F2).

(See Figure 4-13.)








U8 U7 62 U1


U4 /e
U3\4 U5
e3 6e4

U2 65 U6
Figure 4-13


Note that PT(Ul) = PT2(U) = PT2(U)= 1, PT2(U2) = PT2(U3) = PT2(U4) =
PT2(U6) = 2, and PT2(us) =8. Thus we have that m2(1) = 1,m2(2) = 2, m2(8) = 8.

Note that m2(i) does not exist for i = 3,4,5,6, 7. Since ul 0 A(u3) and PT2(U3) =

2, weget F3 = F2U{{u3, us8}} = {el,e2,6 e3,6 e4, e5, e7}. Since the subgraph (V(F3),F3)

is a forest, but not a tree, it is not a maximal tree of G. Define T3 = (V(G), F3).

(See Figure 4-14.)


U8 U7 2 U1

67 61
U3

63 64

U2 6e5 U6
Figure 4-14


Note that PT3(UI) PT3(US) = PT3(U7) = 1 and PT3(U2) = PT3(U3) = PT3(U4) =
PT3(U6) = PT3(Us) = 2. Thus we have that m3(1) = 5 and m3(2) = 4. Note that
mr3(i) does not exist for i = 3,4,..., 8. Since PT3(U4) = 2, we define F4 = F3 U

{{U4, U5}} = {61,6e2,6 e3,6 e4,6 e5,6 e7, e68}. Since V(F4) = {v1,v2,...,vs} = V(G) and
IV(F4)| = 8 = IF41 + 1, the subgraph T = (V(F4), F4) is a maximal tree of G. (See

Figure 4-15.)







US U7 e2 U1
e7 el
U4
U3 5

U2 e5 U6
Figure 4-15

At this moment, we have that PT(Ul) = PT( 2) = ...= PT(us) = 1. Then,
for every ut V(G) with t > 4, we have that Ft = F4. Thus, for every t > 4, the
subgraph (V(Ft), Ft) is same as T = (V(F4), F4).

The next example shows how we get the maximal tree for a given graph
containing some cutvertices.

Example 4.5.2 Let G be the graph in Example 4.2.16. (See Figure 4-1.) Then
CI(G) = {b,c,e} and Cc (G) = {a,d,f,g,h,i,j,k,l,m,n}. In Ci(G), we have that
d(e) = 5, d(b) = 4 and d(c) = 3. On the other hand, in Cc(G), we have that
d(k) = 5, d(m) = d(n) = 3, d(d) = d(f) = d(g) = d(h) = d(l) = 2, d(a) = d(i) =
d(j) = 1. And C2 = 0. Thus we order the vertices as follows; vI = e, v2 = b, v3 = c
and v4 = k, {v5s,v6} = {m,n}, {v7,v8,v9,Vo,vll} = {d,f,g,h,l}, {v12,v13, v14} =
{a,i,j}. Assume that vs = m, v6 = n, v7 = d, V8 = f, v9 = g, vio = h, vi =
1, v12 = a, vi13 = i, v14 = j, which are decided by the sizes and centers of the
corresponding spheres. (See Figure 4-16.)


V5 V11 V13

V6"- V4 Vl0 / V9


V12 V2 V3 V7 V1 V8


Figure 4-16









Since IG(v) = { {v= v}, {v1, vs}, {v1, v9}, {v1, vio}, {v1, v14} }, let F1 = IG(v1).

Then V(F1) = {vI,v7, v8, V9, Vio, V14}. Since V(F1) V(G), the tree T1 = (V(G),F1)

is not a maximal tree of G. Thus, for the cutvertex vi, we have the following sub-

graph which is T1 = (V(G),F1). (See Figure 4-17.)


* V4l


" V4


*12 V2 3
Vl2 V2> V3


V13


V7


V9



V1 V8

V14


Figure 4-17


Since PT,(v2) = 2, and, for every vertex v E AG(v2) = {v3, v4, V6, v12}, PTi(V) 7 1,

let F2 = F1 U Ia(v2). The subgraph (V(F2), F2) is a forest, but not a tree. Thus it

cannot be a maximal tree of G. Now, for the cutvertex v2, we have the following

subgraph which is T2 = (V(G), F2). (See Figure 4-18.)


* VUll


2 V6 V3

V12 V2 V3


V10



V7


Figure 4-18


V9



V1 V8

V14


Since AG(v3) = {v2, v4, v7}, PT (V2) = PT2(V4) = 2 = PT2(V3), and PT2(v7) = 1, let

F3 = F2 U {{v3, V7}}. Then, for the cutvertex v3, we have the following subgraph

which is (V(F3),F3). Then the subgraph (V(F3), F3) is a tree. But V(F3) 5 V(G).








Thus it cannot be a maximal tree of G. We have the following subgraph which is

T3= (V(G),F3). (See Figure 4-19.)


. Vll


V12 V2 V3 V7 Vi V8


Figure 4-19


Since AaG(v4) {v V2, v3,V5, v6, V11}, and

PT3(V2) = PT,(V3) = PT 3(V6) 1 = PT3(V4),PT3(V) =5,PT3(vn) 11,
we define F4 = F3 U {{v4, V5}} U {{V4, v11}}. Then the subgraph (V(F4), F4) is a

tree. But V(F4) 5 V(G). Thus it cannot be a maximal tree of G. Then, for the

cutvertex v4, we have that the following subgraph which is T4 = (V(G), F4). (See

Figure 4-20.)


V5 V11 V13.


V6 V4 V10 V9



V12 V2 V3 V7 V1 V8

V14
Figure 4-20

For the vertices vs, v6, v7, vs, v9, there is no 'new' edge. Thus we get the same sub-

graph as above, that is, T4 = T5 = T6 = T7 = T = Tg. But, for the cutver-

tex vl0, we have that v13 e V(IG(vlo)) and PT,(V13) = 13 $ 1 = PT9(vio). Thus








F10o = F9 U { {vIo,v 13 } and V(Flo) = V(G). Therefore we have the following sub-
graph which is T = (V(G), Flo) = (V(Flo), Flo). (See Figure 4-21.)


V5 V11 V13

V6' V4 Vo1 V9


V12 V2 V3 V7 V1l V8

SV14
Figure 4-21

Clearly, the subgraph T = (V(F1o), Fo10) does not include a cycle. Thus it is a max-
imal tree of G. That is, for every t = 11, 12,13,14, we have that (V(Ft), Ft) =
(V(Flo), Flo). Therefore, the subgraph T = (V(F1o),Flo) = (V(G), Fio) is the max-
imal tree of the graph G.
For this resulting tree T = (V(F1o),Flo), choose a vertex sequence s by
s = (V,V7,V3,v2,V4,V5, Vi,V6,v12,Vlo,V13,Vs,V9,V14). Then the sequence

P = (5,1,1,3,2,0,0,0,0,1,0,0,0,0)
is the degree sequence of A. By Theorem 3.4.12, P satisfies DSC. It is clear that
P also satisfies DSbC. By Corollary 3.6.10, P is the perfect sequence for the tree

T = (V(F1o), Fo).

Theorem 4.5.3 Let G = {V, E} be a graph for a sphere packing plan where
V = {u1, u2,..., un} with the order described in Section 4.4. Let the subgraph
(V(Fn),F,) of G come from the above procedure. Then V(F,) = V(G). There-
fore (V(Fn), Fn) is a maximal tree of G and there is a unique perfect n-sequence for
the tree (V(F,), F,).

Proof The subgraph (V(Fn), Fn) is connected. In fact, assume that the sub-
graph (V(Fn),Fn) contains two disjoint subtrees, namely T, and T2. Then the








path-value for every vertex in T1 is different from the path-value for every ver-
tex in T2. Since the graph G is connected, there is at least one edge e E E(G)
which connects the two subgraphs T1 and T2 and e V E(T1 U T2) = Fn. Let

{ {VJl, Vk }, {vj2, Vk2 },. .. {Vjq, Vkq} } be the collection of such edges where vj, E T1
and vkj C T2 for i = 1,2,. ., q. Without loss of generality, let j, < ji for every
i = 2,3,..., q. Then it is clear that {vjj, Vk} I Fj1 by the procedure of getting the
subgraph (V(Fn), F.). Then {vj,, vk } C F,. This is a contradiction. Therefore the
subgraph (V(F,), Fn) is connected. The path-value in the procedure of getting the
subgraph (V(Fn),Fn) guarantees that the subgraph (V(F,,),Fn) is acyclic. There-
fore (V(F,), Fn) is a tree. It is clear that, for every i = 1,2,... n, the vertex set
V(Fi) contains the vertices u1, u2,...,i u. Thus V(Fn) contains all the vertices of
G, since V(Fn) D {u,u2,...,un} =, V(G). That is, V(Fn) = V = V(G). Hence
the subgraph (V(F,), F,) is a maximal tree of G. By the Tree Classification The-
orem (Theorem 3.6.8), it is clear that there is a perfect n-sequence for the tree
(V(F,), F). I

Definition 4.5.4 The tree shape of a brain tumor is the tree corresponding to that
tumor, or equivalently, its unique corresponding perfect sequence.

Definition 4.5.5 Two tumors have the same shape if and only if their corresponding
graphs obtained from their respective sphere packing plans are the same.

As in Chapter 2, for any brain tumor, we have a unique sphere packing plan
which can be denoted by a graph. From the previous theorem, we have a unique
perfect sequence for the graph. That is, we can assign exactly one perfect sequence to
each brain tumor. So, if two perfect sequences are distinct, then their corresponding
trees, and therefore their respective corresponding graphs and sphere packing, are
also distinct. That is, if two tumors are represented by distinct perfect sequences,
then their corresponding trees are not isomorphic. And their respective graphs and








sphere packing plans are not isomorphic. Thus, we may consider their shapes to be
distinct. Therefore we have the following as our main theorem of this section.

Theorem 4.5.6 The perfect sequences are invariants of the shapes of arbitrary brain
tumors.

4.6 Graph from a Maximal Tree

Let T = {V, E'} be a maximal tree for a graph G = {V, E} which is obtained
from the sphere packing plan of a given tumor or solid shape. We observe that we can
recover the original graph, G, from T. Since V(T) = V = V(G) and E(T) = E' C
E = E(G), we need to add the remaining edges to the edge set of the maximal tree to
get the graph G. Since the graph G comes from the sphere packing treatment plan,
each vertex v in G denotes a sphere of the sphere packing plan. Thus, for every
vertex vi E G, let c, = (xi, y1i, zi) and r, be the center and the radius of the sphere.
By Proposition 4.3.1, for any two vertices vi and vj in V if d(ci, cj) < (r, + r),
then the edge {vi, vj} is in the edge set, E(G), of the graph G. Thus
E'U {{vi,vs} E V x VI d(c2,cj) < "(ri +rj)} = E.
Therefore we get the graph G = {V, E}.

4.7 Algorithm for Our Maximal Tree

Let G = (V, E) be the graph for a sphere packing plan with IV[I = n.
STEP A Find all the i-th step cutvertices of the graph G, for 1 < i < n.
STEP B Label the vertices in the vertex set V by the order explained in Section
4.4.
STEP C By induction, for each i = 1,2,...,n, get the subgraph (V(Fi),Fi) of
G. Then T = (V(Fn),Fn) is our maximal tree.














APPENDIX A
SHELLING PROGRAM FOR SPHERE PACKING
In this appendix, we provide the C++ program we developed to get the sphere
packing plan for a given tumor shape. The first section shows the program translating
the data from CT scan to the data of appropriate form for the main program, which
is in the second section. Note that this program does not include the score function
which was developed by Thomas Wagner. We refer the reader Appendix E for the
complete article on the sphere packing treatment plan.

A.1 Program for Data Reading
// rewrite data file into x,y,z format
// & produce the total # of data, x,y,z min & max

#include
#include
#include
#include
#include

main()
{ const int nofdata=3370; // comes from dataOOS.txt
int xposition[nofdata], yposition[nofdata], zposition[nofdata];
int q;
float a; float b; float c;
int k=0;

{ ifstream inClientFile("data003.txt", ios::in);
while (inClientFile >> a >> b >> c)
{ xposition[k] = int(floor(a)); // check the number
yposition[k] = int(floor(b)); // check the number
zposition[k] = int(floor(c)); // check the number
k++;
}
}
int xmin=xposition[O];
int xmax=xposition[O];
int ymin=yposition[O];
int ymax=yposition[O];
int zmin=zposition[O];
int zmax=zposition[O];

for (int i=O; i { if ( xposition[i] < xmin )
xmin = xposition[i];





83

else if ( xposition[i] > xmax )
xmax = xposition[i];
else if ( yposition[i] < ymin )
ymin = yposition[i];
else if ( yposition[i] > ymax )
ymax = yposition[i];
else if ( zposition[i] < zmin )
zmin = zposition[i];
else if ( zposition[i] > zmax )
zmax = zposition[i];
}
// output the result of centers
{ ofstream outClientFile("mM003.txt", ios::app);
{ outClientFile << nofdata << setw(10) << xmin << setw(5)
<< xmax << setw(5) << ymin << setw(5) << ymax
<< setw(5) << zmin << setw(5) << zmax << endl;
}
}
cout << nofdata << setw(5) << xmin << setw(5) << xmax
<< setw(5) << ymin << setw(5) << ymax << setw(5)
<< zmin << setw(5) << zmax;
cin >> q;

}

A.2 Program for Center Searching
// searching the centers for pat 2

#include
#include
#include
#include
#include

int square(int); // function prototype
int MAX(int,int,int); // function prototype

main()
{ const int nofdata=3370; // check the number from mM003.txt
const int xmin = -32; // check the number
const int xmax = -4; // check the number
const int ymin = -29; // check the number
const int ymax = 4; // check the number
const int zmin = -57; // check the number
const int zmax = -37; // check the number

const int xfix = 1-xmin;
const int yfix = 1-ymin;
const int zfix = 1-zmin;
int xposition[nofdata], yposition[nofdata], zposition[nofdata];





84

int q;
float a; float b; float c;
int k=0;

{ ifstream inClientFile("dataOO3.txt", los::in);
while (inClientFile >> a >> b >> c)
{ xposition[k] = int(floor(a))+xfix;
yposition[k] = int(floor(b))+yfix;
zposition[k] = int(floor(c))+zfix;
k++;
}

const int x = max xmin + 3;
const int y = ymax ymin + 3;
const int z = zmax zmin + 3;
const int nc = x*y*z; // number of cells
int cc[nc]; // status value

for (int i=O; i cc[i] = -3;

int xc[nc]; // x-coord. of c[nc]
int yc[nc]; // y-coord. of c[nc]
int zc[nc]; // z-coord. of c[nc]

// setting the x,y,z-coord. for each cell
for (int i=O; i { zc[i] = floor(i/(y*x));
xc[i] = floor((i-zc[i]*y*x)/x);
yc[i] = i-zc[i]*y*x-xc[i]*x;
}

// setting the boundary of tumor by -1
for (int i=O; i { cc[zposition[i]*x*y + xposition[i]*y + yposition[i]] = -1;
}

// setting the initial of normal by 0
cc[0] = 0;

// setting the normal cell by 0
int h=0;
for (int j=O; j { for (int i=O; i { if ( cc[i] == -3 )
{ if (cc[i-1]==Ol Icc[i+l]==Ol Icc[i-y]==Ol Icc[i+y]==0)
{ cc[i] = 0;
h++;
}
else;
}
else;









if (h==0)
break;
else
h=0;


// setting the remaining tumor cell by -2
for (int i=0; i { if ( cc[i] == -3 ) // checking
cc[i] = -2; // checking
else; // checking
} // checking


// setting the boundary cells
int maxrad;
int nofmaxrad;
for (int j=O; j<=MAX(x,y,z)/2+l; j++)
{ int k=0;
for (int i=O; i { if (cc[i] == -1 II cc[i] == -2)
{ k++:


if ( cc[i-1] == j I
I I cc[i+y]==j
{ cc[i] = j+1;
nofmaxrad = k;


I cc[i+l] ==j 1i cc[i-y]==j
II cc[i-y*x]==j II cc[i+y*x]==j )


else;


else;


maxrad = j;
if (k==0)
break;
else;


Checking the no. of nofmaxrad
Sif nofmaxrad > 100, then you have change the number
S(nofmaxrad >100)
cout << endl << endl << endl << endl << endl <<
<< endl << endl << endl << endl;
cin >> q;


else;



// pick the boxes with maxrad
int suprad[100];
int TotalNofBox4[100];
int TotalNofMax=0;
for (int i=O; i { if (cc[i]==maxrad)








{ suprad[TotalNofMax] =i;
TotalNofMax++;
}
else;
}

cout << TotalNofMax << endl;
cin >> q;


// Count for the number of centers of each case
for (int k=O; k { // reset TotalNofBox4[ I to 0
for (int i=O; i<100; i++)
{ TotalNofBox4[i] = 0;
}
TotalNofBox4[maxrad] ++;


int tcc[nc];
for (int i = 0; i tcc[i] = cc[i];


// remove th<
for (int p=O:
{ int dist


if ( dist
tcc[p]
else;


// temporary cell notion
++i)
// reassign tcc[i] for 'for' loop


a biggest sphere
p Sfloor(sqrt( square(yc[suprad[k]] -yc[p])
+ square (xc [suprad [k] ]-xc [p] )
+ square(zc[suprad[k]]-zc[p]) ));
< maxrad )
= 0;


// setting the remaining tumor cell by -2
for (int i=O; i { if ( tcc[i] >= 1 )
tcc[i] = -2;
else;


// setting the boundary cells
for (int j=O; j<=MAX(x,y,z)/2+l; j++)
{ int 1=0;
for (int i=O; i { if (tcc[i] == -2)
{ if (tcc[i-1] == j I I tcc[i+l] ==j II tcc[i-y]==j
II tcc[i+y]==j II tcc[i-y*x]==j II tcc[i+y*x]==j )
{ tcc[i] = j+1;
++I;
}
else;
}
else;


(1==0)
break;









else;


// check if there is more maxrad pixcel
for (int i=0; i { if ( tcc[i] == maxrad )
{ cout << endl << endl << endl <<
WARNING!!! WE HAVE MORE 'MAXRAD' PIXCEL AT ("
<< xc[i] << "," << yc[i] "," << zc[i] << ")"
<< endl;
cin >> q;
break;


else;


//remove the smaller or = sphere
for (int j=maxrad-l; j>1; --j)
{ for (int i=0; i { if ( tcc[i] ==j )
{ TotalNofBox4[j]++;
// remove the sphere
for (int 1=0; l { int dist = floor(sqrt( square(yc[i]-yc[l])
+ square(xc[i] -xc[1])
+ square(zc[i]-zc[l]) ));
if ( dist < j )
tcc[l] = 0;
else;
}
// setting the remaining tumor cell by -2
for (int i=0; i { if ( tcc[i] >= 1 )
tcc[i] = -2;
else;
}
// setting the boundary cells
for (int j=0; j<=MAX(x,y,z)/2+l; j++)
{ int 1=0;
for (int i=0; i { if (tcc[i] == -2)
{ if ( tcc[i-1] == j II tcc[i+1] ==j ||
tcc[i-y]==j II tcc[i+y]==j II
tcc[i-y*x]==j II tcc[i+y*x]==j )
{ tcc[i] = j+1;
++i;
}
else;
}
else;


(1==0)
break;








else;


else;


output the result of centers
ofstream outClientFile("patOO3-NofCenters.txt", ios::app);
{ outClientFile << endl << endl << endl << endl << endl <<
(k+1) << setw(5) << "(" << (xc[suprad[k]]-xfix) << ","
<< (yc[suprad[k]]-yfix) << "," << (zc[suprad[k]]-zfix)
<< ") with radius = << maxrad << endl;


for (int j=maxrad; j>l; j--)
{ ofstream outClientFile("patOO3-NofCenters.txt", ios::app);
{ outClientFile << setw(4) << j << setw(4)
<< TotalNofBox4[j] << endl;


// Search for centers
for (int k=O; k { int tcc[nc]; // temporary cell notion
for (int i = 0; i tcc[i] = cc[i]; // reassign tcc[i] for 'for' loop

// output the result of centers
{ ofstream outClientFile("patOO3-centers.txt", ios::app);
{ outClientFile << (k+1) << endl << setw(5) << "x"
<< setw(5) << "y" << setw(5) << "z" << setw(10)
<< "radius" << endl << setw(5) << (xc[suprad[k]]-xfix)
<< setw(5) << (yc[suprad[k]]-yfix) << setw(5)
<< (zc[suprad[k]]-zfix) << setw(10) << maxrad << endl;
}
}

// remove the biggest sphere
for (int p=O; p { int dist = floor(sqrt( square(yc[suprad[k]]-yc[p])
+ square(xc[suprad[k]]-xc[p])
+ square(zc[suprad[k]]-zc[p]) ));
if ( dist < maxrad )
tcc[p] = 0;
else;


// setting the remaining tumor cell by -2
for (int i=0; i { if ( tcc[i] >= 1 )
tcc[i] = -2;








else;


// setting the boundary cells
for (int j=O; j<=MAX(x,y,z)/2+l; j++)
{ int 1=0;
for (int i=O; i { if (tcc[i] == -2)
{ if ( tcc[i-1] == j II tcc[i+l] ==j I
II tcc[i+y]==j II tcc[i-y*x]==j II
{ tcc[i] = j+1;
++i;


I tcc[i-y]==j
tcc[i+y*x]==j )


else;


else;
}
if (1==0)
break;
else;


// check if there is more maxrad pixcel
for (int i=O; i { if ( tcc[i] == maxrad )
{ cout << endl << endl << endl <<
WARNING!!! WE HAVE MORE 'MAXRAD' PIXCEL AT ("
<< xc[i] "," << yc[i] << "," << zc[i] << ")" << endl;
cin >> q;
break;
}
else;
}

//remove the smaller or = sphere
for (int j=maxrad-l; j>1; --j)
{ for (int i=0; i { if ( tcc[i] ==j )
{ TotalNofBox4[j]++;
// output the result of centers
{ ofstream outClientFile("patOO3-centers.txt",
ios::app);
{ outClientFile << setw(5) << (xc[i]-xfix)
<< setw(5) <<(yc[i]-yfix) << setw(5)
<< (zc[i]-zfix) << setw(10) << j << endl;
}

// remove the sphere
for (int 1=0; l { int dist = floor(sqrt( square(yc[i]-yc[l])
+ square (xc [i] -xc []l)
+ square (zc[i]-zc[l]) ));
if ( dist < j )
tcc[l] = 0;








else;
}
// setting the remaining tumor cell by -2
for (int i=O; i { if ( tcc[i] >= 1 )
tcc[i] = -2;
else;
}
// setting the boundary cells
for (int j=O; j<=MAX(x,y,z)/2+l; j++)
{ int 1=0;
for (int i=O; i { if (tcc[i] == -2)
{ if ( tcc[i-1] == j I I tcc[i+1] ==j I I
tcc[i-y]==j II tcc[i+y]==j II
tcc[i-y*x]==j I I tcc[i+y*x]==j )
{ tcc[i] = j+1;
++i;
else;
else;


else;


if (1==o)
break;
else;


}
else;
}

ofstream outClientFile("patOO3-centers.txt", ios::app);
{ outClientFile << endl;


int square(int a)
{ return a*a;
}
int MAX(int a, int b, int c)
{ int p=0;
p = a;
if (a<=b)
p = b;
else if (a<=c)
p = c;
else;
return p;














APPENDIX B
PROGRAM FOR THE MAXIMAL TREE OF A GRAPH
In this appendix, we provide the Mathematica program we developed in joint
work with Matthew Harvey. The program below was written by Matthew Harvey to
get the maximal tree for the graph which we assigned for a brain tumor. Note that
this program does not use the method of ordering for the vertices as in Chapter 3.
Instead this program uses some commands which are in the function package in the
Mathematica program library. If anyone wants to develop a program like this using a
different program language, we recommend the vertex ordering algorithm in Chapter
3.
<< DiscreteMath'Combinatorica'
Off[General: :spell]

LexPrecedes[a_, b_] := Module[{i},
For[i = 1, i <= Min[Length[a], Length[b]], i++,
If[ a[[i]] != b[[i]],
If[ a[[i]] > b[[i]], Return[True], Return[False] ] ]
1;
If[ Length[b] > Length[a], Return[False] 1;
Return [True]]

DegreePoint[a_, k_] := Count[Flatten[ToUnorderedPairs[a]], k];

IndSubgraph[Graph[g_, v_], s._List] := Module[{i, j, newmat, sc},
newmat = g;
sc = Complement[Range[Length[v]], s] ;
For[i = 1, i <= Length[g], i++,
For[j = I, j <= Length[g], j++,
If[MemberQ[sc, i] II MemberQ[sc, j], newmat[[i, j]] = 0]]];
Graph[newmat, v]];
CutPoints[gr_] :=
Select[ArticulationVertices[gr], DegreePoint[gr, #] > 2 &]

NonCutPartitions[gr_] := Module[{partitions, cpts, bccs, skeleton},

partitions = {};

bccs = Select[BiconnectedComponents[gr], Length[#] > 2 &] ;

partitions = Union[partitions, bccs];

skeleton =
IndSubgraph[gr,
Union[
Complement[ Range[V[gr]], Flatten[bccs] ],









CutPoints [gr]
]
];

partitions =
Union [partitions,
Select[ConnectedComponents [
FromUnorderedPairs[
Select[ToUnorderedPairs[
skeleton], # !=
Intersection[#,
Flatten[
Select[BiconnectedComponents[skeleton],
Length[#] > 2 &]
]
&]
]
], Length[#] > 1
&I
] ;

Table[
Select[
ToUnorderedPairs[gr],
# == Intersection[ #, partitions[[i]] ]
&], {i, Length[partitions]}]]




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HDPDWD


A CLASSIFICATION OF TREES AND
APPLICATIONS OF TOPOLOGY AND GRAPH THEORY TO
NEUROSURGERY
By
TAE-IL YI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2000

Copyright 2000
by
TAE-IL YI

TO
My parents, Beom-Young & Bong-Wol,
who show me how to love,
My wife, Moon-Sil,
who shows me what is love,
and
My children, David & Peter,
who are the reasons why I love

ACKNOWLEDGEMENTS
I would like to express my sincere appreciation to the members of my supervi¬
sory committee, Dr. Beverly Brechner, Dr. Philip Boyland, Dr. James Keesling, Dr.
Li-Chien Shen, and Dr. Mary Kantowski, for their patience, knowledge, concern, and
especially for showing me how to study and teach. I would especially like to thank
my advisor, Dr. Beverly Brechner, who accepted me and gave me a chance to work
with her. I also want to express my appreciation for all her effort and all that she has
done for me, not only in my major field, but also in my life. I would like to thank
Matthew Harvey who brought me some wonderful ideas and wrote the Mathematica
programs. I would like to thank Dr. Francis Bova who brought me these questions
which were ‘new world’ to me. I would like to thank Dr. Meeks for helpful conver¬
sations, and Dr. Thomas Wagner who gave me the information about neurosurgery
that I did not know before. I would like to thank Dr. Yunmei Chen who gave me
inspiration. I would like to thank Dr. Miklos Bona who showed me some insights of
combinatorics. Finally, I would like to thank the Department of Mathematics that
allowed me to be here and meet all these wonderful people.
IV

TABLE OF CONTENTS
ACKNOWLEDGEMENTS iv
ABSTRACT vii
CHAPTERS
1 INTRODUCTION 1
2 AN AUTOMATED SPHERE PACKING PLAN FOR BRAIN TUMORS 5
2.1 Introduction 5
2.2 Linac 5
2.3 Procedure of Radiosurgery 6
2.4 Making a Sphere by Arcs of Beams 10
2.5 Sphere Packing Plan 10
2.6 Shelling Algorithm 15
3 A CLASSIFICATION OF UNLABELED TREES 25
3.1 Introduction 25
3.2 Definitions 26
3.3 Vertex Sequence 28
3.4 Degree Sequence 36
3.5 Perfect Sequence and Main Theorem 44
3.6 Obtaining Perfect Sequences 45
3.7 Extended Partitions and the Algorithm 53
4 TREES AS INVARIANTS FOR BRAIN TUMOR SHAPES 59
4.1 Introduction 59
4.2 Definitions and Some Properties 60
4.3 Graphs for Sphere Packing Plans 64
4.4 Order in the Vertex Set 65
4.5 Maximal Tree from a Graph 71
4.6 Graph from a Maximal Tree 81
4.7 Algorithm for Our Maximal Tree 81
v

APPENDICES
A SHELLING PROGRAM FOR SPHERE PACKING 82
A.l Program for Data Reading 82
A.2 Program for Center Searching 83
B PROGRAM FOR THE MAXIMAL TREE OF A GRAPH 91
C PROGRAM FOR CLASSIFYING TREES BY SEQUENCES 93
C.l Program 93
C.2 Some Program Results 97
D LABELED GRAPHS AND TREES 102
D.l Number of Labeled Graphs and Labeled Trees 102
D.2 Sketch of the Construction of a Priifer-code 102
D.3 Reconstruction of a Tree from a Priifer-code 103
REFERENCES 105
BIOGRAPHICAL SKETCH 105
vi

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
A CLASSIFICATION OF TREES AND
APPLICATIONS OF TOPOLOGY AND GRAPH THEORY TO
NEUROSURGERY
By
Tae-Il Yi
August 2000
Chairman: Dr. Beverly L. Brechner
Major Department: Mathematics
The work of this dissertation was motivated by our work with a medical group
at the Brain Institute at the University of Florida. This group was led by Dr. Francis
Bova, who raised the following questions:
(1) Can you automate the time-consuming sphere packing process for our “multiple
isocenters method of stereographic radiosurgery” for brain tumors? (This surgery is
noninvasive, radiation surgery to destroy brain tumors.)
(2) Can you classify the shapes of brain tumors?
In this dissertation, we construct some algorithms and programs to (partially)
answer these questions. Our major (joint) results are described below.
Our first result is an algorithm and a program that produce a unique sphere
packing plan for any given brain tumor shape, filling the shape with different size
spheres in a unique way. This process automates the ‘sphere packing’ part of the
Vll

surgery plan. Additional code by the medical group was necessary to allow for physi¬
cian input in certain critical cases. It was then possible for the medical group to
incorporate this into their surgery plan, semi-automating it, and significantly reduc¬
ing the surgeons’ surgery planning time.
Our second result is a complete classification of unlabeled n-trees. We assign
to each n-tree, a sequence of n nonnegative integers satisfying certain properties; and
from each such sequence, we can recover the n-tree assigned to it. In fact, we present
both an algorithm and code to produce not only the sequences, but also pictures of
the corresponding trees.
Our final result is a first attempt at a topological classification of tumor shapes.
We first use our sphere packing plan to construct a graph representing the tumor.
We then have an algorithm and program which uniquely select a maximal tree in the
graph. Thus we can assign a unique tree to each tumor shape. However, different
graphs may produce the same tree. Our tree classification allows us to compare
two tumor shapes very easily, by just looking at their corresponding sequences. In
particular, different sequences come from different sphere packing plans, but the
converse does not hold.
Vlll

CHAPTER 1
INTRODUCTION
The work of this dissertation was motivated by two questions raised by Dr.
Francis Bova, of the Brain Institute, Shands Hospital, and the Department of Neuro¬
surgery, all at the University of Florida. It turns out the different-sounding questions
are very much related and are tied together naturally in this thesis. In fact, our
results for the second question depend upon our results for the first question. Dr.
Bova led a medical group, which joined with our mathematics group, for a number
of discussions over a two year period. This resulted, for us, in both a joint medical
paper [11] and in this dissertation. We also want to reference the dissertation [10] of
Thomas Wagner, Dr. Bova’s medical physics student, who worked with us.
We expect to have three mathematics publications from this dissertation-from
Chapters 2, 3, and 4, respectively. The paper resulting from Chapter 2 will be joint
with Professor Beverly Brechner, Professor Yunmei Chen, and Dr. Thomas Wagner.
The papers resulting from Chapters 3 and 4 will be joint with Matthew Harvey, an
undergraduate student working with Professor James Keesling on his honors thesis.
Further, we anticipate an additional paper, to be joint with Yunmei Chen and her
student, Dr. Stacey Chastain. This anticipated work will be an analysis of the tumor
shapes of approximately 1000 interesting tumor radiation surgeries done by Bova and
his colleagues at the University of Florida.
Dr. Bova’s questions were the following:
1. Is it possible to automate the medical group’s “multiple isocenters method of
stereographic radiosurgery” to destroy brain tumors?
2. Is it possible to classify brain tumor shapes?
1

2
In this dissertation, we develop some algorithms and programs related to these
questions. In stereographic radiosurgery, which is noninvasive surgery, radiation is de¬
livered by means of collimators-cylindrical metal tubes, with cylindrical holes through
them, through which the radiation flows. The goal of stereotactic radiosurgery for a
brain tumor is to deliver the desired dosage to the target, and only the target. This
is not possible in reality. So they do the next best thing, which is to deliver enough
dosage to the target, to avoid as much normal tissue as possible, and to deliver as
little radiation as possible to whatever normal tissue must be affected. There are two
additional important criteria-dose homogeneity and dose conformality. That is, we
do not want ‘hot spots,’ which have been experimentally determined to cause com¬
plications; and we do want rapid falloff of dose levels outside the actual tumor. One
of several such radiation surgery methods is called the ‘Multiple Isocenter Method.’
This involves filling the tumor image with spheres of different sizes, until the image is
best filled up. In this dissertation, we call such a plan, a ‘Sphere Packing Treatment
Plan.’
In Chapter 2, we develop both an algorithm, which is called a ‘shelling process’
(or ‘grassfire process’ in medical terminology), and a C-H- program to identify the
centers and sizes of spheres for this sphere packing treatment plan. For any given
brain tumor, our program gives us a unique sphere packing plan. This sphere packing
plan can be represented numerically as a list of parameters of the spheres, namely
the centers and radii of the spheres. Based on this algorithm, the medical group
developed a proprietary program for an automated sphere packing treatment plan,
including dosage distribution, and other things (see Wagner et al. [11]). Since this
program always produces the same plan for the same tumor, it can be used as a
‘benchmark’ against which to compare other treatment plans.
In Chapter 3, we obtain a code to denote an unlabeled tree by a unique,
ordered, nonnegative integer sequence satisfying certain properties. We call such

3
sequences, perfect sequences. That is, we obtain a complete classification of unlabeled
n-trees by means of the perfect sequences of n nonnegative integers. Thus we have our
main theorem of this chapter, the Tree Classification Theorem, (Theorem 3.5.5): For
any positive integer n, let T(n) be the set of unlabeled n-trees and F(n) the set of
perfect sequences of n-trees. Then there is a one-to-one correspondence between T{n)
and V{n). We show how to assign a perfect sequence to a given tree, and conversely,
how to reconstruct a tree from a given perfect sequence. We also include an algorithm
and a Mathematica program to produce not only all the perfect n-sequences, for any
given positive integer n > 3, but also pictures of the corresponding trees. By means of
this program, we get our classification of unlabeled n-trees and also the total number
of n-trees.
In Chapter 4, we make a first attempt at a topological shape classification of
tumors-actually of the sphere packing plan associated with the tumor shape. We first
assign to each tumor a unique graph, and from that, we determine a unique tree. We
obtain the graph by assigning a vertex for each sphere, and an edge between each pair
of vertices representing adjacent spheres. This produces a unique connected graph
corresponding to each sphere packing plan. Then, by using the notion of cutvertex, we
give an order to the vertex set. We then use this order to build a unique maximal tree
contained in the graph. Next, we develop an algorithm and a Mathematica program
to assign a unique tree to each sphere packing plan. This program produces a list of
edges of the tree, making it clear which edges are incident.
Using our results of Chapter 3, we can now assign exactly one perfect sequence
to each tumor. However, it may happen that more than one tumor shape (or graph)
gives rise to the same tree, and therefore to the same perfect sequence. Thus, we do
not yet have a real classification of tumor shapes. However, if two perfect sequences
are distinct, then their corresponding trees, and therefore their respective correspond¬
ing graphs and sphere packing plans, are also distinct. Therefore we may consider

4
their shapes to be distinct. Thus, we have our main result of Chapter 4, Theorem
4.5.6: The perfect sequences are invariants of the shapes of arbitrary brain tumors.
That is, if two brain tumors are represented by distinct perfect sequences, then their
corresponding trees are not isomorphic, and therefore their respective graphs and
sphere packing plans are not isomorphic. Thus, we may consider their shapes to be
distinct.
We refer the reader to Douglas[1], Gross[3], and Harary[5] for graph theory.
We also refer the reader to Hall[4], Riordan[7] and Stanley[9] for combinatorics.

CHAPTER 2
AN AUTOMATED SPHERE PACKING PLAN FOR BRAIN TUMORS
2.1 Introduction
The goal of stereotactic radiosurgery for a brain tumor is to deliver the desired
dosage to the target, and only the target. This is not possible in reality. So they do
the next best thing, which is to deliver enough dosage to the target, to avoid as much
normal tissue as possible, and to deliver as little radiation as possible to whatever
normal tissue must be affected. There are two additional important criteria-dose
homogeneity and dose conformality. That is, we do not want ‘hot spots,’ which
have been experimentally determined to cause complications; and we do want rapid
falloff of dose levels outside the actual tumor. One of several such radiation surgery
methods is called the ‘Multiple Isocenter Method.’ This involves filling the tumor
image with spheres of different sizes, until the image is best filled up. This noninvasive
method of surgery, namely by using radiation, relies on a piece of equipment called the
Linear Accelerator (or simply, Linac). In this chapter, we show how to automate the
construction of a sphere packing plan for the tumor image. Most of the information
in this chapter about treatment of a brain tumor is taken from Friedman et al. [2].
See [2] for further reference.
2.2 Linac
According to Friedman et al. [2], the linear accelerator is a complex machine
capable of producing X-rays. A large amount of energy is generated by the power
supply, which then powers the filament shown. This causes electrons to be emitted
by the filament, which are in turn accelerated to higher energies using a (microwave)
5

6
wave guide. The electrons are then changed in direction by the magnet so that they
impact on a heavy metal alloy target. This results in X-ray production that can then
be collimated or shaped by both primary and secondary collimators within the linear
accelerator head. This beam is further collimated for radiosurgery by the tertiary
radiosurgery collimator. (See Figure 2-1 and Figure 2-2.)
The Linac is mounted on a rotating gantry such that the beam has a center of
rotation about 1.5m above the floor. Usually, the isocenter accuracy is defined within
a 2mm sphere. Because stereotactic radiosurgery depends on optimized accuracy, an
improved system was designed at the University of Florida, by adding a set of bearings
to the stereotactic collimator system and under the patient table. As a result, this
new system achieves mechanical accuracy within 0.2mm±0.1mm for defining the
treatment isocenter of beam delivery.
The tertiary collimators are generally circular and allow improved centering
of the treatment beam. The sizes of these collimators are from 5 to 40mm in 2- to
5-mm increments.
2.3 Procedure of Radiosurgery
The procedure of radiosurgery is separated into 5 steps as follows.
(1) Head ring attachment
(2) CT scan
(3) Image Processing
(4) Planning
(5) Treatment delivery
For more detailed information, refer to Friedman et al.[2].
(1) Head ring attachment
The stereotactic radiosurgery treatment requires attachment of a stereotactic
head ring. This ring provides accurate information about the position and the shape

Filament Waveguide
Electron beam
Power supply
\.
ti II
•—H
Whr
Secondary (Adjustable)
collimators
Tertiary (Radiosurgery)
collimators
Photon beam tor
radiosurgery
Floor stand with treatment arm above
Figure 2-1. System of Linac (Friedman et al.[2])

8
Figure 2-2. Operation of Linac (Friedman et al.[2])

9
of brain tumors from the computer tomography (CT). This ring also helps to insure
that the treatment target is accurately placed at the precise isocenter of the Linac.
(2) CT scan
After attaching the head ring, a series of cut filming is taken. Eighty to 150
CT slices are taken per patient. For radiosurgical treatment planning, getting the
accurate size and shape of the target is very important. Underestimation of the target
size may result in treatment failure. Overestimation of size results in the inclusion of
normal brain tissue within the treatment volume. Misrepresentation of an irregular
target shape may lead to radiation damage of normal brain tissue. The accuracy
of CT scanning depends on the size of the pixels that make up the scan image.
The lateral and AP accuracy of scanning is approximately 0.6mm. But the axial
dimension is determined by slice thickness of the CT scan, which is 1mm in general.
The physicians use the so-called Brown-Roberts-Wells coordinates-or, simply BRW
coordinates-which are called lateral (Lat), anterior-posterior (AP), and axial (Ax)
coordinates. These are the x, y, and z coordinates, respectively. The direction of
the lateral coordinate is from the left hand side ear to the right hand side ear of the
patient. So the lateral view (or sagittal) is similar to the side view of the head, that
is, views of slices along the lateral axis. This is the same as yz-plane views. The
direction of the AP coordinate is from the back side of the head to the face. So the
AP view (or coronal) is similar with the front view of the head, that is, views of slices
along the AP axis. This is the same as xz-plane views. The direction of the axial (or
vertical) coordinate is from the neck to the top of the head. So the axial view (or
trans-axial) is similar to the sky view of the head, that is, views of slices along the
axial axis. This is the same as xy-plane views. (See Figure 2-4.)
(3) Image Processing
An MRI scan has been done the day before surgery. Using a mouselike device,
the CT and MRI images are transferred to the computer screen and fused. Then,

10
using geometric equations, the computer determines the lateral, AP, and vertical
stereotactic coordinates of each point in each CT slice.
(4) Planning
Physicians use a mouse and their own judgment to ‘eyeball’ the location and
the size of spheres for the treatment plan.
(5) Treatment delivery
The geometric sphere packing plan, developed as in (4), is combined with a
proprietary dose planning program, and the resulting treatment plan is delivered by
the Linac.
2.4 Making a Sphere by Arcs of Beams
By varying the angle of the gantry and the angle of the table, one can deliver
a radiation beam to the target from any angle within the range of the rotation. The
shape of the common intersection of an arc of beams passing through one isocenter is
a sphere. The neurosurgeons deliver a series of arcs (usually 5 or 9 arcs) to produce
a single isocentered sphere shape. For an ellipsoidal target, they use fewer numbers
of arcs to make a single isocentered ellipsoid shape. (See Figure 2-3, 2-4 and 2-5.)
So, if the target shape is very close to a sphere or an ellipsoid, then the
treatment plan is relatively easy compared to an irregularly shaped target. In that
case, we need to create a geometric treatment plan.
2.5 Sphere Packing Plan
As seen in the previous section, the physicians know how to irradiate-to-
(eventually)destroy tumors which are shaped like spheres or ellipsoids. For a non-
spherical shape of tumor, they try to fill the target with several spheres of different
sizes. This is called the ‘sphere packing’ treatment plan.
After finding a sphere packing plan, they treat each sphere separately as de¬
scribed in the previous section. So, multiple isocenter radiosurgery planning includes

11
Figure 2-3. One isocenter plan with five arcs of radiation (Wagner et al.[ll])

12
(270*)
Figure 2-4. One isocenter plan with nine arcs of radiation (Friedman et al.[2])
111 Q U CQ

13
TARGET
Figure 2-5. Plan for a sphere shape and an ellipsoidal shape (Friedman et al.[2])

14
the problem of determining the best sphere packing arrangement with which to fill
the target volume. General methods for this treatment plan are iteratively based,
dosimetrically driven algorithms. But these methods require many computations in
order to compute a radiosurgical plan dose distribution, and then to evaluate the
quality of the dose distribution. So geometrically based radiosurgery optimization
has been suggested as a possible alternative means.
However the method the physicians choose relies on human decisions and
experience. Thus, for the same target, different surgeons may produce different plans.
Even the same surgeon, doing the plan twice for the same target, may produce
different plans. And the planning takes a long time, especially for a complicated
target which needs more than 10 spheres. It might take as much as two hours of
planning for a difficult case which needs about 20 spheres. During that time, the
patient has to wait with the head ring attached to his or her head. And most
importantly, even after spending the time to make a plan, many physicians without
sufficient experience, are not sure if the plan is a ‘good’ one.
Therefore, we provide an automated sphere packing method for the treatment
plan (see Wagner et al.[ll]). This method shows potential to significantly aid the
planning of difficult multiple isocenter cases. Based on tests with irregularly shaped
phantom targets and with a representative sampling of clinical example cases, the
method demonstrates the ability to generate radiosurgery plans comparable to, or
of better quality than, multiple isocenter Linac radiosurgery plans found in other
literature. At the same time, this program always produces the same treatment plan
for the same tumor shape. So it can be used as a ‘benchmark’ to compare with other
plans for the given tumor shape. Moreover, this program provides the treatment
plan in a relatively short time. For a very difficult case which needed more than 18
spheres, this program took less than 3 minutes instead of more than the 1.5 hours

15
which were needed when the physicians created the plan using traditional methods.
The algorithm of this program is shown in Figure 2-6.
In the following section, we explain the ‘shelling algorithm’ which we used in
this program to get the centers and sizes of spheres for the sphere packing plan.
2.6 Shelling Algorithm
The shelling procedure is best illustrated in Figure 2-7 to 2-16. The major
steps of this shelling procedure are as follows.
(1) Transferring the data of the boundary of the target
From the CT scan, each point which is identified from the target contour data
file can be mapped into a three dimensional array, since each CT scan shows the
trans-axial view for a specific axial coordinate value. In the three dimensional array,
we assign the lateral-, AP- and axial-coordinates to be x-, y- and z-axes, respectively.
For each individual case, there are minimum and maximum values for x, y and z-
axes. We assume the minimum value is always one. We get this information from
the Program for Data Reading. (See Appendix A.l.)
Let the target be minimally enclosed in an x-range beginning with one and
ending with Mx, a y-range beginning with one and ending with My, and a z-range
beginning with one and ending with Mz. Now we put ‘safe bumper’ layers sur¬
rounding the target. That is, we enlarge the box containing the target to ET =
[0, Mx +1] x [0, My +1] x [0, Mz + 1]. ET stands for ‘enlarged target box.’ We assume
that the dimension of each voxel is lxlxl. Then we will number the voxels in a lexi¬
cographic manner, beginning with 0 and ending with (Mx +1) x (My + 1) x(M* + l) — 1,
giving us a total of (Mx + 1) x (My + 1) x (Mz + 1) voxels with which we will work.
Now we want to identify the voxels in ET in two different ways: first, by a set
of coordinates (a, 6, c), and second, by an integer representing a counting procedure.
Here, a is the left hand endpoint of the x-value of that voxel, b is the left hand

16
endpoint of the y-value of that voxel, and c is the left hand endpoint of the z-
value of that voxel. The second notation for the voxel (a, 6, c) is by some integer
n. The voxels are counted, using a lexicographic ordering beginning with 0 and
ending with (Mx + 1) x (My -f 1) x (Mz + 1) — 1 which is denoted by nofdata in
the program (see Appendix A.2). Then the voxel denoted by (a, 6, c) is numbered
as c x (Mx + 1) x (My + 1) -f a x (My + 1) + b. And it is clear that, for every
0 < i < nofdata, there are unique nonnegative integers, a, 6, c, such that i —
c x (Mx + 1) x (My + 1) + a x (My + 1) + b. That is, for every 0 < i < nofdata,
one and only one voxel is assigned. Note that Figure 2-7 to Figure 2-14 show the
same trans-axial view at different stages of the algorithm. Thus these figures show
the data of a fixed axial-coordinate value.
(2) Assign a status value for each voxel
For every 0 < i < nofdata, the status value for the z-th voxel is denoted
by cc[i\. That is, we denote the status value for the voxel with coordinates (a, 6, c),
by cc[i\ where i = c x (Mr + 1) x (My + 1) + a x (My + 1) + b. The following is
the procedure to assign status values. First of all, let cc[i\ = —3 for every 0 <
i < nofdata. Then we assign the number, —1, as the status value for each voxel
corresponding to any of the target contour data. Next, for every 0 < j < Mz, we
assign cc[j x (Mx + 1) x (My -J-1)] = 0, because the j x (Mx + 1) x (My + l)-th voxel
is the voxel at the left lower corner for every j-th trans-axial view; that is, the voxel
belongs to the healthy tissue. (See Figure 2-8.)
For each voxel with cc[i\ = —3, we define cc[i\ — 0 if any of cc[i — 1], cc[i +
1], cc[i — (My + 1)], cc[i + (My + 1)] is 0. Note that this method produces zeros for
every voxel in the safe bumper layers. (See Figure 2-9.)
(3) Shelling the target voxels
We proceed by mathematical induction on j, for
0 - x min{Mx, My, Mz}

17
For i = 0 to (Mx + 1) x (My -f 1) x (Mz + 1) - 1, if the i-th voxel is assigned -1
or —3 for its status value, and if any of cc[i — 1], cc[i + 1], cc[i — (My + 1)], cc[i +
(My + 1)], cc[i - (Mx + 1) x (My + 1)], cc[t + (Mx + 1) x (My + 1)] is equal to j,
then we assign a new status value for the ¿-th voxel as c\i\ = j + 1. Note that Figure
2-7 to Figure 2-14 show only a two-dimensional figure. But this procedure is held
in a three-dimensional array. This process is repeated until the deepest lying voxels
have been identified. The deepest lying voxel in the entire target volume would be
the best location for an isocenter. And at the same time, the status value assigned to
the deepest voxel indicates the radius of the sphere, which is one less than the status
value of the voxel. This completes our induction. (See Figure 2-10.)
(4) Removing the largest sphere
At this moment, we might have several different voxels which have the same
largest status value. So we choose the smallest numbered voxel as the center of the
first largest sphere to be removed from the target. But in the program in Wagner et
al. [11], we choose the most effective voxel to be the center of the first largest sphere,
by using the ‘score function’ which was developed by Tom Wagner. We think of
the resulting sphere in the tumor as ‘removed,’ and therefore we can assign the new
status value zero for these voxels in the sphere. To determine which voxels form the
sphere that is removed, we do the following: Let R = voxel at the center of sphere
to be removed. Let r = status value of R. Then we remove all voxels S with the
property that distance(center of R, center of S) < r — 1. (See Figures 2-11 and 2-12.)
Now, we proceed by mathematical induction on j, for
0 - x min{Mx, My, Mz}
For i = 0 to (Mx + 1) x (My + 1) x (Mz + 1) — 1, if the ¿-th voxel has status value
> j, we reassign a new status value as follows: if any of cc[i — 1], cc[i + 1], cc[i —
(My + 1)], cc[i + (My + 1)], cc[i-(Mx + l) x (My + 1)], cc[i + (Mx + 1) x (My + 1)] is
equal to j, then we assign a new status value for the ¿-th voxel as c[¿] = j + 1. This

18
process is repeated until the deepest lying voxels have been identified. (See Figures
2-13 to 2-15.)
And we repeat step (4) until we get either 0 or 1 for the status value of each
voxel in ET, the enlarged target box. Because the smallest size of the diameter of
collimators is 5mm, we finish our program when the status values of all the voxels are
zeros and/or ones. (See Figure 2-16.) The remaining part will be taken care of by
the ‘score function’ (see Wagner et al.[ 11 ]). After finishing this procedure, we have
the spheres filling the target. In Figure 2-17, we see the sphere packing plan within
a target by this method.
For further detailed information about the score function and the automated
sphere packing plan, we refer the reader to Wagner et al.[11] and Wagner[10].
Figure 2-6. Sphere packing algorithm block diagram (Wagner et al.[11])

19
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Figure 2-7. Transferring the data of the boundary of the target
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Figure 2-8. Status values for the boundary and the initial voxel

20
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Figure 2-9. Status values for the normal voxels
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~cT
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Figure 2-10. Shelling the target voxels

21
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Figure 2-11. Choosing the largest sphere
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Figure 2-12. Removing the largest sphere

22
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Figure 2-13. Shelling the remaining target voxels
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Figure 2-14. Choosing the largest sphere of the remaining target

23
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Figure 2-15. Removing the second sphere
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Figure 2-16. Removing all the spheres

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Figure 2-17. A sphere packing plan for a brain tumor (Wagner et al.[ll])

CHAPTER 3
A CLASSIFICATION OF UNLABELED TREES
3.1 Introduction
In this chapter, we obtain a code to denote an unlabeled tree. By means
of this code, we classify unlabeled n-trees. One may take an unlabeled n-tree to
be the geometric realization of a connected, finite, simplicial complex without any
simple closed curves. In particular, we call a tree an n-tree if and only if it is a tree
with n vertices (that is, (n — 1) edges). Our code assigns a unique, ordered, ‘perfect
sequence’, pf(T) = (cfi, d2,..., dn), to each unlabeled n-tree, T. And, conversely,
given an ordered sequence of n integers satisfying certain properties, it is the perfect
sequence of exactly one unlabeled n-tree. Thus, we have our main theorem, the Tree
Classification Theorem (Theorem 3.5.5). We then show how to reconstruct the tree
shape from a given perfect sequence, by hand.
Our work includes an algorithm and a Mathematica program (see Appendix
C.l) that produce a list of all the perfect sequences for all possible n-trees, thus also
producing the number of n-trees, for any given n. For some examples, see Appendix
C.2. Since our algorithm and program are based upon our main theorem, we know
that all possible unlabeled n-trees are produced by the program. Thus the program
gives us, for each positive integer n, a complete list of all unlabeled n-trees.
Given n > 0, with sufficient computer power and time, our program will
list all the unlabeled n-trees, thus telling us how many unlabeled n-trees there are.
However, we do not have a simple formula that tells us how many unlabeled n-trees
there are. This remains an open problem.
25

26
The Literature contains work regarding the classification of labeled graphs
and labeled trees. In Lovász[6], the Priifer-code is used to determine the number of
labeled n-trees (see the Appendix D.2 and D.3). In Sloane[8], Sloane shows a table
for n < 26. However, Sloane references Harary for the method he uses. In Harary[5],
there is a simple formula for the total number of labeled graphs with n vertices (see the
Appendix D.l). There is also a formula for the total number of unlabeled n-trees, but
it requires knowing the number of rooted n-trees, which is calculated inductively. And,
even though Harary[5] contains n-tree shapes up to n = 10, there is no explanation
of how to get the shapes of these trees, while our work does show how to get these
tree shapes from our code.
This chapter is organized as follows:
Section 1: Introduction
Section 2: Definitions
Section 3: Vertex Sequence
Section 4: Degree Sequence
Section 5: Perfect Sequence and Main Theorem
Section 6: Obtaining Perfect Sequences
Section 7: Extended Partition and the Algorithm
3.2 Definitions
Most of the necessary definitions in this section are from Harary[5]. We refer
the reader to Harary[5] for easy reference.
Definition 3.2.1 A graph G = (V(G), E(G)) consists of a vertex set V(G) =
{t>i, and an edge set E(G) = {e i,..., em} C {{u, u}| u,v G V(G)} in which
each edge is an unordered pair of two distinct vertices u and u, and the edge {u,u}
is said to join u and v. For an edge {u,u} we say that u and v are adjacent
vertices and the vertex u and the edge {u,u} are incident with each other, as are

27
v and {u,v}. From now on, |V(G)| denotes the number of vertices and |E(G)\
denotes the number of edges. For a vertex u € F(G), the vertex set
A{u) = {u £ G\v is an adjacent vertex of u}
is called the adjacent set of u. Note that u # A(u).
Definition 3.2.2 A walk is an alternating sequence (i>0, ei, v\, e2,..., e*, Vk) of
vertices and edges s.t. u,_i / ut- for i — 1,2, and e,- = {u,_i,n,} for i =
1,2,..., fc. If there is no confusion, a walk can be denoted by a sequence of vertices
only in the path like (iq, v2,..., Vk) . A cycle is a walk (n0, ei, iq, e2,..., e*, n*) with
A: > 3 in which Vo = Vk is the only vertex repeated. A path is a walk with no
repeated vertex. A (u,v)-path is a path with the first vertex u and the last vertex
v. The length of the (u,u)-path, l(u,v), in a graph is the number of edges in the
(u,i>)-path. The distance d(u,v) between two vertices u and v is the length of a
shortest path joining them, if any; otherwise d(u,v) — oo. Note that d(u,u) = 0. A
graph G is connected iff every pair of vertices are joined by a path. A graph having
no cycle is acyclic. Note that an acyclic graph can be disconnected. A forest is an
acyclic graph. A tree is a connected acyclic graph. An n-tree is a tree with n vertices.
Definition 3.2.3 We say that a graph G is isomorphic to a graph H if and only
if there exists a bijection / : V{G) —> V(H) such that {u,i>} £ E(G) if and only if
{Hu)J(v)}€E(H).
Definition 3.2.4 An unlabeled tree is an isomorphism class of trees.
We are interested in the total number of unlabeled n-trees and their shapes.
For example, Tx = ({1,2,3}, {{1,2}, {2,3}}) and T2 = ({1,2,3}, {{2,1}, {1,3}})
are different trees, but there is an isomorphism / : V'(Tj) -> V(T2) by /(1) =
2, /(2) = 1, /(3) = 3. Thus they are the same unlabeled tree, so we want to count
them as one unlabeled 3-tree. From now on, a tree implies an unlabeled tree unless
otherwise specified.

28
Definition 3.2.5 The degree of a vertex v, defined by d(v), is the sum of the
number of edges incident with v. A vertex v is called an endvertex if d{v) — 1.
We will use the geometric, that is, topological realization of a graph inter¬
changeably with ‘graph’ as defined above.
Proposition 3.2.6 [5] The following statements are equivalent for a graph T :
(1) T is a tree.
(2) Every two vertices of T are joined by a unique path.
(3) T is connected and |V(T)| = \E(T)\ + 1.
(4) T is acyclic and |V(T)| = |¿£(T)| + 1.
Note that there are exactly n vertices and n-1 edges in an n-tree.
Proposition 3.2.7 (Degree sum formula) [5] If G is a graph with the vertex
degrees du. then E„GV(G) = 2\E(G)\-
Corollary 3.2.8 For an n-Tree T we have
ET=i d{vi) = 2(n - 1) = 2\E(T)\ = 2(\V(T)\ - 1)
where is a vertex of T for i = 1,2,..., n.
Because a nontrivial tree satisfies Ylvev(T)d(v) — 2\E(T)\ — 2(\V(T)\ — 1),
there are at least two vertices with degree 1. Thus we have the following theorem as
a corollary.
Theorem 3.2.9 [5] Every nontrivial tree has at least two endvertices.
3.3 Vertex Sequence
In this section, we define the vertex sequence of a given n-tree.
Definition 3.3.1 Let T = (V, E) be an n-tree. If d(y) = m, then there are m
components of T — {u}. A branch of v (or simply a v-branch) is a subtree which
is the closure of a component of T — {u}. We call the set of vertices in a u-branch
a v-branch vertex set. If u 6 A(u), then the w-branch vertex set containing u is
called the (v,u)-branch vertex set and denoted by B(v,u) (or B(u)). We call the

29
set B'(v,u) = B(v,u) — {u} the (t>,ii)-subbranch vertex set, which is denoted by
B\u) if there is no confusion. The subtree consisting of B'(v,u) as the vertex set
and {{s,t} e B\v,u) x B'(v,u)| {s,t} € E(T)} as the edge set is called the (i>,u)-
subbranch.
Example 3.3.2 Let T be the 10-tree shown below.
s
t
u
p
<7
r
V
w
X
y
Figure 3-1
Since d(v) = 4, there are 4 components of T — {v}. Since A(v) = {r, t, to, y},
we have that
B{v,r) = {v,r,s,q,p},
B(v,t) = {M},
B(v,w) — {v, io, u, a:},
B{v,y) = {v,y},
B'(v,r) = {r,s,q,p}
B'(v,t) = {¿}
B'(v, w) = {w, u,
B'(v,y) = {y}.
On the other hand, we have that
B(r,v) = {r,v,t,u,w,x,y}.
That is, B(v, r) / B(r, v). And this is true for any tree except the 2-tree. In fact, for
every n-tree with n > 2, there are at least 3 vertices, namely u,v, and w. Without
loss of generality, let u and v be vertices adjacent to each other, and let the (io, v)-
path contain the vertex u. Then w £ B(v,u) and w ^ B(u,v).
Definition 3.3.3 A section of a sequence is a subsequence consisting of consecutive
terms of the sequence.
Definition 3.3.4 Let T = (V, E) be an n-tree with V = {aq, X2,..., xn}. We define
the set, S, of sequences of the form s = (tq, V2,..., vn), where s is a permutation
of the Xi s in V(T), to be those sequences satisfying:

30
(*) for any Vj with 1 < j < n, Vj+\ € A(vk) — {ni, v2,..., Uj}
where k = max{i € {1,2,... ,j}| A(v{) — {vi,v2,... ,Vj} / 0}.
Then S is called the vertex sequence set of T and each element, s, in S is called a
vertex sequence of T.
Definition 3.3.5 For any v € V, we call Sv = € 5|ui = v} C S the v-vertex
sequence set.
Definition 3.3.4 requires that the vertices in a sequence be listed in a particular
way. The condition (*) ensures that all vertices are listed and that each is listed
exactly once. We can choose any vertex for the first term, iq. Then the condition
(*) says that, for the (i+l)-th term, we have to choose an unlisted adjacent vertex
of V{, if possible. If every adjacent vertex of u,- is already used, then look backward
along the sequence until we come to the first vertex which has an unused adjacent
vertex. Then, for the (i+l)-th term, we assign one of the unused adjacent vertices of
that vertex. To get a vertex sequence, we continue this procedure inductively.
Note that every vertex sequence of an n-tree has n entries.
Example 3.3.6 We show how to construct a vertex sequence of a given tree. Let
T be the 7-tree shown below.
f 9
a b c d e
Figure 3-2
Since we have freedom to choose the first term, let V\ = b. Then v2 € í4(üi) —
{ui} = A(b) — {6} = {a, c, /} — {6} = {a, c, /} = A(b) = A(ui). So v2 must be one of
the elements of A(ui) = A(b) = {a,c, /}. Let v2 = c. Since v3 € A(v2) — {t>i,u2} =
A{c) — {b, c} = {b, d, g} — {b, c} = {d,g}, v3 must be either d or g. Let v3 = g.
Then v4 # A(v3) - {vi,v2,v3}, since A(v3) - {vuv2,v3} = {c} - {b,c,g} = 0. On

31
the other hand, we have that A(v2) — {t>i,n2, u3} = {&, d,g} — {b,c,g} — {d} / 0.
Therefore we have to choose d as u4. Then v^ — e since A(v4) — {iq,v2,u3,14} =
{c, e} — {b,c,g,d} = {e}. Since we have
A(u5) - {vi,V2,V3,v4,v5} = {d} - {b,c,g,d,e} =0,
A(n4) - {vi,v2,v3, v4, u5} = {c, e} - {6, c,p, d, e} =0,
A(n3) - {ui,n2,n3,n4,n5} = {c} - {&, c,g, d, e] =0,
A(u2) - {ui,n2,u3,u4,n5} = {M, d} - {&, d, e] =0, and
A(üi) - {vi,v2,u3,u4,u5} = {a, c, /} - {6, c,g, d, e] ={a,f},
ve must be either a or /. Let ve = /. Then we have that
A{vG) - {vi,v2,v3,v4,v5,ve} = {&} - {b,c,g,d,e,f} =0,
A{v^) - {vuv2,v^vA,vh,vG} = {d} - {b,c,g,d,e,f} =0,
A(v4) - {v1,v2,v3,v4,v5,ve} = {c,e} - {b,c,g,d,e,f} =0,
A(u3) — {ui, v2, t73, u4,175, v6} = {c} - {b,c,g,d,e,f} =0,
A(v2)-{vi,u2,u3,u4,u5,u6} = {6,sr, rf} - {¿>,c,£,d,e,/} =0, and
A(ni) - {vuv2,v3,v4,v5,v6} = {a, c, /} - {&, c,£, d, e, /} = {a}.
Therefore iq must be a; that is, we construct a vertex sequence
(iq,u2,...,iq) = (b,c,g,d,e,f,a).
For a given tree we will get different vertex sequences if we choose different
vertices whenever we have two or more choices. But the sequence t = (6, c, d, e, /, a, i/)
cannot be a vertex sequence of T above. In fact, we have v5 = f. But we have that
A(u4) - {iq,u2,u3,u4} = {d} - {b,c,d,e} =0,
A(v3) - {vuv2,v3,v4} = {c, e} - {6, c, d, e} =0,
A{v2) - {tq, u2, t>3, 14} = {6, d,5-} - {6, c, d, e} = {#}, and
4(ui) - {iq,n2,u3,u4} = {a, c, /} — {6, c, d, e} ={a,/}.
That is, the sequence t has /, one of the elements of A(rq) — {iq, n2, v3, n4}, for U5
instead of g which is an element of A(v2) — {iq, n2, u3, tq}. It is a contradiction to
the maximality of k in the definition of vertex sequence.
Proposition 3.3.7 Let (iq, t>2,..., vn) be a vertex sequence of an n-tree T. For any
Vj with j > 1 there is unique 1 < k < j such that Vj G A(vk) — , v2,..., tq_4}.
Proof First of all, for any j > 1, we have Vj / n, for i = 1,2,... ,j — 1 from the
Definition 3.3.4. Assume that there is j > 1 such that
{Vj,Vj+ i,...,U„} D ((Uf¿i4(ü<)) U {vi,V2,...,Vj_i}) = 0.

32
Then, for any u £ {vj,vj+i,...,un} and any w £ (U¿_j A(n¿)) U {rq, u2, ■.., Vj},
there is no (u,u;)-path in T. That is, T is not connected. It is a contradiction
to the connectedness of a tree. Therefore there is l < k < j such that vj €
A(vk) — {ui, u2, ■ ■ ■, Vj-i}. If there are 1 < k\ < k2 < j such that Vj £ A(vkl) —
{ui,u2,...,Uj_i} and Vj £ A(vki) - {ui, v2,.. ., u¿-i}, then the two edges, {vj,vkl}
and {vj,vk2}i are elements of E(T). There is a path between vkl and vk2 in the
subtree {{iq, v2,..., Uj_i}, E'} where E' = {{u, u} £ E(T)\u, v £ {ui, u2,..., *b_i}}.
The tree T contains a cycle consisting of {vj,vkl}, {vj,vk2} and the (vkl, t?fc2)-path.
This contradicts the fact that a tree is acyclic. Therefore there is only one l < k < j
such that vj £ A(vk) — {i>i, v2,..., Uj_x}. T
From the Definition 3.3.4 and the previous lemma we get the following corol¬
lary immediately.
Corollary 3.3.8 Let (ui, u2,..., vn) be a vertex sequence of an n-tree. Then
(1) v2 £
(2) If d(vj) > 2 for any j = 2,3,..., n — 1, then Vj+1 £ A{vj).
(3) For any j = 2,3,..., n — 1 if d(vj) = 1 and there is i < j such that d(vf) > 3
and d(vk) = 2 for every k = i + 1, i + 2,... ,j — 1, then uJ+1 £ A(v{) and Vj+i £
A(vp), for every p = 1,2,... ,i - l,i + 1,..., j.
Proposition 3.3.9 Let (ui,u2,... ,vn) be a vertex sequence of an n-tree and A(ui) =
... ,ulm} with 1 < li < 12 < ... < lm. Then
{v1,v2,...,vn} = U”L1£(v1,ult).
Proposition 3.3.10 Let s = (t?x, u2,..., vn) be a vertex sequence of a given n-tree
T. If d(vj) = m for some j > 1, then there exist m vertices, Vj,Vj1,Vj2t... ,Vjm_l
such that A(vj) = {u¿, vjl, vj3,..., ujm_1} with i < j < ji < j2 < ... < jm-1 •
Proof For any j > 1 there exists unique i < j such that {u,-,Uj} £ E(T)
by Proposition 3.3.7; that is, £ A(vj). So, for any vk £ A{vj) — {u,}, j < k.

33
Since d(vj) = m, there are m — 1 vertices, Vj1,Vj2:.. • such that A(vj) —
{Vi, Vh ,vh,..., ujm_1} with i < j < ji < ji < ... < jm-1 • -L
Proposition 3.3.11 Let s = (v\, u2, â–  â–  â– , vn) be a vertex sequence of a given n-tree
T. Let d(vi) = m and A(ui) = {uin vl2,..., uim} be the adjacent set of vu named
so that 1 < li < 12 < ... < lro- Then, for any i = 1,2,..., m, the set of vertices in
the subbranch, B'(vi,ui¿), forms a section in s starting with v\t.
Proof Clearly, the section containing Ui, starts with Vil = u2. Let B'(yi,V\f) =
{ui1? Uij+i,.. •, }. Assume that there is a vertex u G V(T) — B\vi, UiJ such
that (uijjUij+i,... ,ui 1+j,«,...) is a section of s. Since u £ B'(v i, Vi,), A(vll+k)~
B'{vuVll) = 0 for any k = 0,1,2, ...,j from the definition. Thus U^=0A(ui1+a;) —
B'(v!,«!,) = 0. That is, U^.=0A(ui1+fc) D B'(vijUiJ. On the other hand, A(ui1+¿) C
B'(v i,uij) forevery k = 0,1,2,..., j. Thus we have that U^.=0A(ull+fc) C B'(vi, ui,).
Therefore U^._0yl(ui1+fc) = B'(vThat is, j = mi. Hence B'{vi,ui,) forms
a section in s. Since B'{yi,^,) is a subbranch of v\, Ui1+TOl+i 0 A(u) for any
u e B'(vi,vij). Thus ull+mi+1 G A(vi). That is, ui1+mi+1 = vl2. For any i =
1,2, ...,m — 1, let ¿T(di,Ui.) form a section in s and Ui,-+mi+i = By a simi¬
lar method, B'(yi,Uil+1) forms a section in s and Uii+1+rn,-+1+i — nit+2. -L
The section formed by a subbranch B'(yi,ui.) is denoted by (B'(vi,Ui¿)).
From the previous proposition, we have the following corollaries.
Corollary 3.3.12 Let s = (ui, u2,..., vn) be a vertex sequence of a given n-tree
T. Let d(vi) = m and A{vi) = V\2,..., Uim} be the adjacent set of Ui, named
so that 1 < lx < 12 < ... < lm. The section (B'(vi,fp)) precedes the section
(B'(v i,vlj+l)) in s for i — 1,2, ...,m — 1, and there is no other vertex between
these two sections.
Corollary 3.3.13 Let (ui, u2,..., vn) be a vertex sequence of a given n-tree T.
For any j > 1 with d(vj) = m > 2 let A{vf) — {ut, Vjl,Vj2,..., Vjm_1} where i <

34
j < ji < ji < ••• < jm—i • Then (B'ivjiVjt)) = (vjt, vji+1,..., %+1)-i) for any
i = 1,2,..., m — 1. In particular, we have that j\ = j + 1.
Note that = {iq, v¿m_uV¿m_i+i,..., uJm-i} is the (iq, )-branch
vertex set for some jm > jm-1-
Corollary 3.3.14 Let (iq, u2,..., vn) be a vertex sequence of a given n-tree. If
i < j with vj € j4(i>,) and Vj £ A(vk) for every k = i + 1, i + 2,..., j — 1, then
vp £ A(vk) for every p > j and k = i + 1, i + 2,..., j — 1.
For any given vertex sequence s = (vi,U2,..., vn) and any vertex Vj ^ tq, v\
belongs to a branch of vj. We want to give a special name to this branch.
Definition 3.3.15 Let s = (iq, v2> • • ■, vn) be a vertex sequence of a given n-tree
T. For any j > 1 the root branch of Vj in s is the branch of Vj containing iq.
Note that there is no root branch for iq.
Proposition 3.3.16 Let (iq, u2,..., vn) 6 S be the vertex sequence set of a given
n-tree T. For any 1 < j < n, if iq+i ^ then vj is an end vertex.
Proof If Vj+1 ^ A[vj) for some 1 < j < n, then vj has a unique edge with a
vertex, u¿, with i < j by Proposition 3.3.7. That is, d(vj) — 1. Therefore Vj is an
end vertex. X
Proposition 3.3.17 Let (tq, u2,..., vn) € S be a vertex sequence set of a given
n-tree T. For any 1 < j < n there is an integer jm such that
{ui, u2, • • •, Vj} U {vjm,vjm+l,..., vn}
is the vertex set of the root branch of Vj.
Proof Let l < j < n and d(vj) = m. Then, by Proposition 3.3.10, there exist
m different vertices, u¿, Vj,, tq2,..., Vjm_l such that A{vj) = , Vj,, ,..., Vjm_l}
with i < j < ji < j-2 < ... < jm-1- From Corollary 3.3.13, we have that
(Ufc=i Tl (vj,Vjk)^ — (i>jj, Ujj+i,..., vjm-i).

35
That is, the union of all the subbranches of tq except the root branch is a section
from Ujj to Therefore the set of vertices in the root branch is {tq, tq,..., tq}U
{Vjm1Vjm+ -L
Note that, if jm > n, then the root branch is {tq, tq,..., tq}.
Corollary 3.3.18 Let s = (tq, tq,..., vn) 6 S be the vertex sequence set of a given
n-tree T. Let j / 1. Then d(vj) = 1, that is, Vj is an end vertex of T, if and only
if the set of vertices of the root branch of Vj is V(T) = {tq, tq, • • •, fn}.
Proposition 3.3.19 Let s = (tq, tq,..., vn) be a vertex sequence of a given n-
tree T. For any vertex vj with j > 1 the (vi,vj)-path in T is a subsequence of
(Vi,V2,...,Vj).
Proof For any vertex Vj with j > 1 there is a unique vertex Vkp with kv < j
such that Vj € A(vkP)- For the vertex Vkp there is another unique vertex Vkp_,
with A;p_i < kp such that Vkp € A(vkp_i). Then we will have that iq is the unique
adjacent vertex of a previous vertex â–  Therefore the (tq, Uj)-path is a subsequence
(vi,Vkl,...,Vkp_l,Vkp,Vj) of (vi,v2,...,Vj). 1
Proposition 3.3.20 For any two vertex sequences ¿j and s2 in the v-vertex se¬
quence set Sv = {s € *Sjvi = v} the root branch vertex sets of a vertex x ^ v\ — v
in si and in s2 are same.
Proof Let S\ = (tq, v2,..., vn) and s2 = (iq, u2,..., un) be two different vertex
sequences of a given tree T with tq = u\. Let x be an arbitrary vertex in T
with x iq. Then there are two integers k and / such that tq = x = u¡. Let
A(x) = {x\,x2,... ,xm-\,t/}. Since T is a tree, there is a unique (tq,x)-path in T.
Then one of the elements of A{x) is in the (tq, x)-path. Without loss of generality,
let y be in the (tq, a:)-path. From the proof of Proposition 3.3.17 we have that
(uJLV B'(x, xr)) = Vk-\-• • • 5 Vk+r}
(U^i'B'tx.x,,)) = (U ^B'(u,,xr))
— (Ul+11 Ul+2, ■ • • 1 Ul+r)

36
for some r. Therefore we have that
{iq, ^2? • • • , Vk} U {Ufc-j-r+l , • • • , tyi} — {^1, ^2, • • • , M/} U {«/+r+l, • • • , ^n}’
That is, the root branch vertex sets of x in sj and in s2 are same. J_
Example 3.3.21 Let T be an 8-tree as shown below.
v w
p q r s t u
Figure 3-3
Clearly, Si = (q,r,s,t,u,w,v,p), s2 = (q,p,r,s,w,t,u,v) e Sr = {s €
Sjiq = q}. Let
(vi,v2,... ,v8) = {q,r,s,t,u,w,v,p) and (iq, u2,..., u8) = (q,p,r,s,w,t,u,v).
Then the root branch vertex set of s is {p, q, r, v}. That is, {p,q,r,v} = {q, r} U
{v,p} = {ri,u2} U {v7,u8} and {p,q,r,v} = { Therefore the root branch vertex sets are same but the position might be different.
On the other hand, d(w) = 1 and the unique adjacent vertex of w — v&
is s = V3. Thus s = V3 is in the root branch of w = ve. Therefore the root
branch vertex set of w = v§ is {iq, v2, V3, U4, v5, u6} U {u7, u8} = {tq, v2,..., u8} =
{q,r,s,t,u,w,v,p}. That is, the root branch vertex set of an end vertex is V’(T)
itself.
3.4 Degree Sequence
In this section, we define the degree sequence of a vertex sequence.
Definition 3.4.1 Let S be the vertex sequence set of an n-tree T. For any ver¬
tex sequence s = (tq, v2,..., vn) € S, we call Da = (di, d2,..., dn) the degree se¬
quence of s where cfi = d{vj) and di = d(v{) — 1 for i > 2. We call the set
D = € 5} the degree sequence set of S. And, for any vertex v € V, we

37
call Dv — {D¡\s € S',,} C D the v-degree sequence set of Sv. For any section
(Bfavjt)) = (vj,vji,vji+u...,vji+l-i) of s, formed by the (vj, UjJ-branch vertex
set B(vj,vJt), we define Bd(vj,vjt) = {dj, dj{, dji+i,..., dJi+1-i} as the (vj, vJt)-
branch degree set where d^ — d(vk) — 1 for k = j, ji, ji + 1, ■ ■ ■, ji+i — 1. And we define
the (vj,Vj^-branch degree section by (Bd(vj, Vjt)) = {dj,djiidji+1,..., dj1+1-i) which
is denoted by (Bd(vj{)) if there is no confusion. In the same way as in Definition
3.3.1, we define {Bd'(vjt)) = (Bd'(vj,vh)) = (djitdji+1,... A+1-i) .
Example 3.4.2 Occasionally, it may happen that there are two or more vertex se¬
quences of a given tree with the same degree sequence. In the following example, there
are several pairs of vertex sequences with the same degree sequences, respectively.
p r s i
Figure 3-4
Since there are 5 vertices in V(T), there are exactly 5! different seqnences of vertices
of T having length 5 and using each vertex only once. But only 16 of them satisfy
the condition (*) in Definition 3.3.4. Therefore the vertex sequence set, S, of the
given tree, T, consists of the following sequences
«1 = (P,r,q,s,t), s7 = (r,p,9,5,i), s\3 = (s,r,p,q,t),
¿2 = (p,r,s,t,q), s8 = (r,p,s,t,q), s\4 = {s,r,q,p,t),
¿3 = (q,r,p,s,t), s9 = (r,q,p,s,t), sis = {s,t,r,p,q),
s4 = (q,r,s,t,p), sIo = (r,q,s,t,p), s\6 = (s,t,r,q,p),
Ss = (t,s,r,p,q), six = (r,s,t,p,q),
s6 = (t, s, r, q, p), sl2 = (r, s, t, q, p).
Then the degree sequences of these vertex sequences are
Afi = <1,2,0,1,0), 0* = <3,0,0,1,0), Dsh =<2,2,0,0,0),
Ds-2 = (1,2,1,0,0), D3-s = <3,0,1,0,0), DaU =<2,2,0,0,0),
Ds-3 = (1,2,0,1,0), Ds-9 = (3,0,0,1,0), A¡ís = (2,0,2,0,0),
A4 = (1,2,1,0,0), Aio = (3,0,1,0,0), Af« = (2,0,2,0,0),
As = (l,l,2,0,0), Dsh =(3,1,0,0,0),
Dse = (1,1,2,0,0), Dah = (3,1,0,0,0).

38
Now we have that
Ds\ = Ds-3,
D=D-
U Sg — U s 10 5
Ds‘i — D s\ i
Dsh ^S12)
Dss — Ds~6 ,
Ds{4 i
Ds[3
D s7 — Ds-g,
Dsi 5 D¡¡i 6
That is, for some different vertex sequences, the degree sequences might be same.
And, if the degree sequences are same, then the trees denoted by these degree se¬
quences are isomorphic.
Remark: The sum of the terms in any degree sequence of a given n-tree is the
same, namely, n — 1. It is 4 in the above example, and it is same as the number of
edges in the tree. See Theorem 3.4.6 below. Since d\ = d(vi) and d, = d(u,) — 1 for
i = 2,3,..., n, we have the following proposition.
Proposition 3.4.3 Let D¿ — (¿i, ¿2, • • •, dn) be a degree sequence of a vertex se¬
quence s of an n-tree T. Then 0 < d\ < n —1, and 0 < dt < n — 1 for i = 2,... ,n.
Proposition 3.4.4 If dj ^ 0 for any 1 < j < n, then Vj+\ is an adjacent vertex
of Vj.
Proposition 3.4.5 Let s = (uj, u2,..., vn) be a vertex sequence of an n-tree T and
D¡ = (di, d2,..., d„) the degree sequence of s. For any j € {1,2,..., n — 1} with
dj — m > 1, if the set {it¿+1, Vj+2, •.., Vk} is the collection of all the vj-subbranches
except the root subbranch, then k > j + m.
Proof Since dj = m = d(vj) — 1, from Proposition 3.3.10, the vertex Vj has m
subbranches besides the root subbranch. Thus there are at least m endvertices in
the collection of those subbranches other than the root branch. Therefore at least m
terms in (dj, dj+i,..., dk) are 0. Hence k > j + m. J_
Theorem 3.4.6 (First Reduced Degree Sum Formula) For any degree sequence
Ds = (di, d2,..., dn) of a vertex sequence s (E S of a given n-tree T we have
ELi di = n-\ = \E\
and
+ EIUW — 1) = 0.

39
Proof d\ is the degree of a vertex, and the remaining n — 1 terms are each one
less than the degree of a vertex. By the degree sum formula (Proposition 3.2.7), it is
true that ]C"=1 d(w¿) = 2\E\. So
£”= i di = di + di
= d\v i) + E"=a(d(t,0-1)
= d(vl) + £"= 2 d(vi) - £”=2 1
= + £IU «M) -(«-1)
= £”=i ¿(«O -("-1)
= 2|i?| - (n - 1)
= 2(n — 1) — (n — 1)
= n — 1 = |£|.
Then we have that
di + ET=2(* - !) = + £"=2 di “ £"=2 1
= £?=i^ -(»-i)
= (n- 1) - (n- 1)
= 0.1
Corollary 3.4.7 For any degree sequence, D~s = (di, d2,..., dn) of a vertex sequence
s € S of a given n-tree T, we have
di + £¿=2, d,jto(di — 1) = £,-2, ¿i =o 1-
Proof From the first reduced degree sum formula we have that di + £tn_2(d,' — 1) = 0.
Then
0 = d\ + £"=2(d,- — 1)
= £"_2i d,jto(di ~ 1) + £"=2, d,=o(di - 1)
= dl + C=2, d,/oK' - 1) + £"=2,
= d\ 1 £,—2| di¿o(di — 1) — £¿=2, d,=o !•
Thus we get the result. T
Note that di > 0 for any i = 1,2,..., n. So d, — 1 < 0 if and only if di = 0.
Corollary 3.4.7 shows that, in a degree sequence D (efi, d2,..., dn), the total
number of 0 terms is the same as d\ + X^”=2 (di — 1). Therefore there are exactly
d,/0
* + £[ l_2 d,^o(di ~ 1) endvertices in the given tree. Because of this, we have the
following proposition.
Proposition 3.4.8 For any degree sequence = (d1? d2,..., dn) of a vertex se¬
quence s 6 S of a given n-tree T, we have

40
di + E¿=2(d¿ ~ 1) 7^ 0
/or euen/ 1 < q < n.
Proof Assume that there is q < n such that d\ + — 1) = 0. Since D¡ =
(¿i, ¿2,dn) is a degree sequence of a vertex sequence s £ S of the given n-tree
T, we have that vq+i £ A(up) for some p < q. Thus at least one of the endvertices
counted by vp is one of vq+i, vg+2,..., vn. That is, in the section (di: d?,..., dq),
we have at least one less 0 terms than d\ + 0(d¿ — 1)? that is,
d\ +
EU
di¿ o(d«
1)-
EU, di
=0i
= di
+ EL
z2(di — 1)
- di
+ E?=
:2, d,¿o(di
-1)
+ EU,
d,= 0
= di
+ EL
=2, di¿o(di
-1)
+ EU,
di =0
= di
+ EL
=2, di¿o(di
-1)
- EU
d,=0
> 1.
This is a contradiction. Therefore we have d\ + ^¿_2(dt- — 1)^0 for every 1 < q <
n. X
Theorem 3.4.9 (Second Reduced Degree Sum Formula) Let T — (V, E)
be an n-tree. Let D¡ = (d\, ¿2, • ••, dn) be the degree sequence of a vertex sequence
s £ S of the given n-tree T. For any jk > 1, let B'[vjk) — üj*+i,uj*+i-i}
be a (vj,Vjk)-subbranch vertex set. Then we have
4+- a=o
and
djk T ELj*+i(d¿ — 1) 7^ 0
for every jk < q < jk+l.
Proof Let E{B'(vjk)) = {{u,o} £ E(T)\u,v £ B'(vjk)}. Then
%) = (%j,£(%j))cr
is also a tree, and (B'(vjk)) = (vjk, Vjk+i,..., %+1-i) is a vertex sequence of the
tree T(vjk). Note that d(vjk) = djk +1 in T. Since Vj $ B'(vjk), the edge {vj,Vjk}
is not in the edge set E(B’(vjk)). Thus we have that d(vjk) = djk in the subtree

41
T(vjk). Then the result comes from the first reduced degree sum formula (Theorem
3.4.6) and Proposition 3.4.8. JL
Corollary 3.4.10 Let s = (tq, i?2, â–  â– ., vn) be a vertex sequence of an n-tree T and
Ds — (¿1,^2, • • • ,dn) the degree sequence of s. For any dj > 0, if the subsequence
(vj, Vj+i,..., vp) is the collection of all the Vj-branches except the root subbranch of
Vj, then
dj + Y7i=j+i(di - 1) = 0
and
di + HLj+Adi ~ 1) ± 0
for every j < q < p.
Proof For j — 1, they are true by the first reduced degree sum formula (Theorem
3.4.6) and Proposition 3.4.8. Let j > 1 and Vj 6 A(vt) for some t < j.
(i) If dj = 0, then d(vj) = 1, that is, vj is an end vertex. Since j > 1, Vj v
Thus the root branch is the whole tree by Corollary 3.3.18. Therefore (Vj) is the
collection of all the Uj-branches except the root subbranch and dj + Y^=J+i(di — 1) =
dj = 0.
(ii) If dj > 0, then the subsequence (vj, Vj+1,..., vp) which is the collection of all
the Uj-branches except the root subbranch, is the (u¿, Uj)-subbranch, since Vj € A{vf).
By the previous theorem, we get dj + J2Pi=:+i(di — 1) = 0. Thus the collection of all
the Uj-branches except the root subbranch of Vj is the (vt, Uj)-subbranch. Therefore,
by the second reduced degree sum formula, we get dj + YH=j+i(di — 1) / 0 for every
j < q < p. 1
Not all of the nonnegative integer sequences with n entries can be a vertex
sequence of an n-tree. There is a condition to being a degree sequence of an n-tree.
Definition 3.4.11 Let D — (ofi, d2, • • •, dn) be a nonnegative integer sequence.
We say that D satisfies the degree sequence condition, briefly DSC, if and only if it
satisfies all of the following:

42
(a) di > 0.
(b) £"=1^ =n-h
(c) For any j < n with dj = m > 0, there are exactly m + 1 integers, j < ji =
j + 1 < ji < • • • < jrn < jm+1(= n + 1 if j = 1), such that
4 + YÁkJjl+\(di - 1) = 0 for k = 1,2
(d) For any jk with djk = l > 0, there are exactly l + 1 integers, jk < jk, =
jk + l < jk2 < • • • < jk, < ifc(I+1) = jfc+i such that
+ 71(rf¡-i)= 0 for p = 1,2,...,/.
Theorem 3.4.12 Leí D — (dj, d2? • • •, ^n) a nonnegative integer sequence. Then
D is a degree sequence of a vertex sequence of an n-tree if and only if D satisfies
DSC.
Proof [=>•] This follows from Proposition 3.4.3, Theorem 3.4.6 and Theorem 3.4.9.
[<=] Let V = {ui,ü2, ... ,u„} be a set of n different vertices. Let D —
(di, o?25 • • •, dn) be a nonnegative integer sequence satisfying the above conditions.
Since di > 0 and d¿ > 0 for any i = 2,3,..., n, we can assign the degree of each
vertex ut- 6 V by d{y\) = d\ and d(v{) = d, + 1 for i = 2,3,..., n. Let j < n and
dj = m > 0. From (c), there are exactly m + 1 integers, j < ji < j? < ... < jm <
jrn+1, such that ji = j + 1, and djk + £¿=j‘+l(d,- - 1) = 0 for k = 1,2,..., m. Now
we assign an edge between vj and vjk for k = 1,2,..., m. That is, Vjk € A(vj)
for i — 1,2,..., m.
(i) If djk = l > 0, we assign an edge between Vjk and Vjkp for p = 1,2,...,/
by (d). We do a similar procedure for each vertex, Vj, assigned to dj > 0. Therefore
vjk e A(vj) and vjkp (ii) If dj{ = 0, then we do not assign any new edge for the vertex vJt. But it is
already joined to the vertex Vj by an edge.
Let s = (v\,V2,..., vn). Then all the vertices in s are connected. Since each vertex
Vi is assigned di new edges and £"_! d¿ = n — 1 from (b), we have n vertices and

43
n — 1 edges. Then the set of all the edges assigned above, together with the vertex
set V, forms an n-tree by Proposition 3.2.6. _L
Remark: The proof of Theorem 3.4.12 shows how to construct a tree that corre¬
spond to the given degree sequence satisfying DSC.
Example 3.4.13 Let D = (3,2,3,1,0,0,0,1,0,2,0, 0,0) be a nonnegative integer
sequence having 13 entries. Let V — {wi,t>2,... ,vn} be a set of n different vertices.
^9
fn
.t>13
, V8
v7
VlO
1^1
V2
V3 V4 V
Vu v6
Figure 3-5
(a) Clearly, = 12 and d\ = 3 > 0. So we expect a 13-tree.
(b) Since d(yi) — 3, there are 4 different integers, 1 < li = 2 < I2 = 10 < I3 =
13 < I4 = 14, such that d2 + £f=3(d¿ -1) = 2 +(2+ 0-1-1-14-0-1) =
0, d\Q + — 1) = 2 + (—1 — 1) = 0, and ¿13 = 0. We assign edges between
Vi and each of u2,i>i0, U13.
(c) (i) Since v2 = 2, there 3 different integers, 2 < 3 < 8 < 10, such that ¿3 +
Yl]=4{di — 1) = 3 + (0 — 1 — 1 — 1) = 0, and d$ + (dg — 1) = 1 — 1 = 0. We assign
edges between v2 and each of v% and u8.
(ii) Since u10 = 2, there are 3 different integers, 10 < 11 < 12 < 13, such that
dn = 0 and ¿12 = 0. We assign edges between r>io and each of t>n and Vi2.
(iii) Since vi3 — 0, we do not assign any new edge for U13. But the vertex V13
is connected to iq by an edge.

44
In a similar way, we get {v3, v4}, {i>3, u6}, {u3, u7}, {u8,U9}, and {n4,n5}. Let E be
the collection of all the edges above, that is,
E = {{ui, u2}, {*>1, vio}, {vu t>i3}> {^2, u3}, {u2, us}, {uio, Un},
{uio, U12}, {u3, U4}, {u3, U6}, {u3, u7}, {u8, u9}, {u4, u5}}-
Since the edge set E contains 12 elements, the set T — {V, E} is a 13-tree.
3.5 Perfect Sequence and Main Theorem
In the previous section, we showed that, for a given n-tree T, there exist many
vertex sequences depending on the starting vertex and choice among the adjacent
vertices for each successive vertex. Therefore the degree sequence set D for the
given tree T contains many different degree sequences which denote the same tree
T. We need a way to choose one degree sequence representing the tree T. First of
all we want to define an order on the set of all finite, nonnegative, integer sequences.
Definition 3.5.1 For any two finite nonnegative integer sequences D = (pi,p2, ■ •■iPk)
and E — (qi,q2, , D < E if and only if either there is m < min{A;,/} such
that pm < qm and pi — qi for all i = 1,2,..., m — 1, or k < l and = q¡ for all
i = 1,2,..., k. We write D < E if D < E or D — E.
Proposition 3.5.2 Let D be the set of all degree sequences of an n-tree. The degree
sequence set, (D, <), is a totally ordered finite set.
Proof For any two different degree sequences the order defined above is well defined.
Since each term dt of a degree sequence is nonnegative and less than n and the
length of each degree sequence is n, the degree sequence set D is a finite set. Thus
(D, <) is a totally ordered finite set. _L
Definition 3.5.3 For a given n-tree T, let S be the vertex sequence set of T and
D the degree sequence set of T. Then the maximum degree sequence in D is called
the perfect sequence of the given tree T, and it is denoted by pf(T). For the perfect
sequence, pf(T), of a given tree, T, there is a vertex sequence s = (iq, u2,..., vn) €

45
S such that the degree sequence of 5 is pf{T). We call s a perfect vertex sequence
of the tree and V\ a perfect vertex of the tree.
In the Example 3.4.2, the perfect sequence is (3,1,0,0,0} and there are two
different perfect vertex sequences, sh = (r,s,t,p,q) and s¡2 = (r,s,t,q,p) for it.
But the trees represented by Sn and S12 are isomorphic.
For any given tree the perfect sequence is unique by Proposition 3.5.2. But
there may exist several different perfect vertex sequences. If the given tree is a
labeled tree, then these perfect vertex sequences denote different labeled trees. But
these labeled trees are isomorphic to each other; that is, there is a unique unlabeled
tree. Therefore, for a given perfect sequence, there exists a unique unlabeled tree.
Theorem 3.5.4 Let d¡ be the perfect sequence of a tree T. If T' is any other tree
whose perfect sequence is also d$, then T' is isomorphic to T.
Proof If T has more than one vertex sequence with the same degree sequence, then
these vertex sequences all produce the isomorphic trees, by using the construction in
the proof of Theorem 3.4.12. T
Because of this theorem, we have our main theorem below.
Theorem 3.5.5 (Tree Classification Theorem) For any positive integer n, let
T(n) be the set of unlabeled n-trees and V(n) the set of perfect sequences of n-trees.
Then there is a one-to-one correspondence between T(n) and V(n).
3.6 Obtaining Perfect Sequences
By Theorem 3.5.5, if we can construct the set of perfect sequences of n-trees,
P(n), without being given any tree shape, then we can classify all the n-trees by this
set. And, in the same way as in Example 3.4.13, we can get the different tree shapes
from the elements of V(n).
Proposition 3.6.1 For the perfect vertex iq of a perfect vertex sequence s =
(tq, v2,..., vn), we have d{yi) > d(vi) for ¿ = 2,3, ...,n.

46
Proof If there is some k € {2,3, ...,n} such that d{v > d{v i), then D-s =
(d(vi),...) < Dy = (d(vk), ■ ■ .) for any vertex sequence s' = (tq,...). This contra¬
dicts the maximality of D-s. T
Proposition 3.6.2 If D-s — (di, d2,..., dn) is the perfect sequence of a given n-tree
T, then n — 1 > d\ > d¿ > dn — 0 for any i = 2,3,..., n — 1.
Proof Since T is an n-tree and D-s is the perfect sequence, we have n — 1 > d\. For
any vertex u £ V(T) we have d(iq) > d(u) by Proposition 3.6.1. Since d\ = d{vi)
and d{ = d(ut) — 1 for every 1 < i < n, we have d\ > di for 1 < i < n. Since dn
must denote one of the end vertices, each of which has degree 1, we have dn = 0.
Therefore we have dt > dn = 0 for every 1 < i < n — 1. T
Definition 3.6.3 Let D = {dl5 d2,..., dn) be a degree sequence. Then we say that
D satisfies the ordered subbranch condition, denoted OSbC, if and only if D satisfies
the following condition for all j > 1 :
If d(üj) = m + 1 > 2, and S'(uJ2), ..., B'(vjm) with j < ji < j2 <
... < jm are the subbranches of the vertex Vj other than the root subbranch, then
(Bd’(vjk)) > (Bd'(vjk+1)) for every k = 1,2,..., m — 1.
Recall that (Bd'^Vjf)) is a section of the degree sequence D. (See Definition
3.3.3 and 3.4.1.)
Remark : If j = 1, then there is no root branch of tq; instead, there is one more
subbranch, Bd'(vjm+l), such that jm < jm+1 and {B'(vjrn)) > {Bd’(vjrn+1)) .
Theorem 3.6.4 Let D = (di, d2,..., dn) be a nonnegative integer sequence. If D
is the perfect sequence of an n-tree, then it satisfies the DSC and the OSbC.
Proof It is clear from Theorem 3.4.12 and the maximality of D. _L
Proposition 3.6.5 Let T be a tree and v a vertex with the largest degree in T.
Then there is exactly one degree sequence Dv for T starting at v and satisfying
both DSC and the OSbC.

47
By the above proposition, the converse of Theorem 3.6.4 is not true. The
following example is a counterexample.
Example 3.6.6 Let A = (cfi, d2:..., d\2) = (3,2,2,1,0,0,2,0, 0,1,0, 0). Then
E,=i ¿i = 11 and dx = 3 > 0. And the sequence A can be separated into 4
sections like (3), (2,2,1,0,0,2,0,0) , (1,0), (0). The second section can be sep¬
arated into 3 sections, (2), (2,1,0,0), (2,0,0), and the second and third sec¬
tion of these can be separated by (2), (1,0), (0) and (2), (0), (0), respec¬
tively. All of these subsections satisfy the second and third conditions of DSC.
Thus the sequence A satisfies the DSC. That is, the sequence A is a degree se¬
quence of a vertex sequence (iq, v2,..., V12) of a 12-tree by Theorem 3.4.12. On the
other hand, the sequence A satisfies the OSbC, since we have (2,2,1,0, 0,2,0,0) >
(1,0) > (0). But A is not the perfect sequence of the 12-tree. In fact, we rearrange
the sequence A to get B = (d7 + 1, d2, di — 1, di0, dn, di2, d3, d4, d5, d&, dg, dg) =
(3, 2,2,1,0,0,2,1,0,0,0,0) . Note that d(rq) = d\ and d(vj) — ¿7 + 1. Clearly the
sequence B also satisfies the DSC and the OSbC. So B is also a degree sequence
of another vertex sequence (tq, v2, rq, Uio, t>n, Vu, V3, tq, u5, vq, v$, vq) of the 12-tree.
That is, A and B are degree sequences of the same 12-tree (see Figure 3-6). And
we have
B = (3,2,2,1,0,0,2,1,0,0,0,0) > (3,2,2,1,0,0,2,0,0,1,0,0) = A.
Thus A is not the perfect sequence. Furthermore A cannot be the perfect sequence
of any tree, because A and B denote isomorphic trees, and B > A.
vn iq0 iq U12

48
In this example we have that d(vi) = 3 = d(v7) which is the largest degree
in the tree. That is, the perfect sequence must start with the largest degree. We will
see in Example 3.6.8 that the degree sequence B is the perfect sequence of this tree.
Note that (1,0) is the only possible degree sequence when d\ — 1. Therefore
we assume d\ > 1 from now on.
Now we show how to construct alternate degree sequences from A, similar
to B. Assume that we have a nonnegative integer sequence A = (cfi, d2, â–  â–  ., dn)
satisfying the DSC and the OSbC. If there is 1 < i < n such that d\ < d,-, then
A cannot be the perfect sequence of any tree by Proposition 3.6.2. So we assume
that d\ > d{ for i — 2,3, ...,n, and there is at least one integer 1 < j < n such
that dj — di — 1. That is, there is an n-tree T = (V, E) such that A is a degree
sequence of a vertex sequence s = (rq, V2,..., vn) of T, and T has at least two
vertices of maximum degree d\. Clearly, dj > 0. Thus Vj is not an end vertex in
T. By Corollary 3.3.18, the set of the vertices of the root branch of Vj is not the
set of vertices in A itself. So, from Proposition 3.3.17 there is j < m < n such
that RB(vj) = {uj, v2> • • •, Vj} U {uTO+i, um+2, • • •, vn} is the set of the vertices of
the root branch of vj. Clearly, (wj+i, u,-+2,..., vm) is the section containing all of
the subbranches of Vj except the root branch. Therefore, if dj — p, then there are
p subsections of A like (Bd'^Vj^) > (Bd'(vj2)) > ... > (5d'(uJp)) with j < j\ <
J2 < ... < jp < m +1, since A satisfies OSbC. On the other hand, let E(RB'(vj)) =
{{u,u} £ E(T)\u,v £ RB'(vj)} be the set of edges in the root subbranch oí Vj. Then
V = {RB'(vj), E(RB'(vj))} is a subtree of T. Let S(T') be the vertex sequence
set of T'. By Proposition 3.3.7, there is a unique integer 1 < k < j such that
Vk € A{vj). Clearly, Vk € RB'(vj). Let SVk(T') be the v^-vertex sequence set of
S(T'), in which the first term of each vertex sequence is u*.. Let DVk(T') be the
Vk-degree sequence set of SVk(T') (see Definition 3.4.1). Since DVk{T') is a finite
set, there is a maximum sequence in DVk(T'). We denote the maximum sequence in

49
DVk(T') by mdR(vk). Note that di must be changed to d\ — 1 in mdR(vk), since
it is not going to be used for the first term anymore in the new degree sequence.
Now we have that either mdR(vk) > (Bd'(vjx)), or there is an integer 1 < q < p
such that (Bd'(vjq)) > mdR(vk) > (Bd'(vjq+ J). Note that mdR(vk) may be the
smallest section, then Vj., would not exist.
We rearrange the given degree sequence A to a new sequence in which dj + 1
is the first term and the subbranch degree sequences of Vj,
mdR(vk), (Bd'(vh)) , (Bd'(vh)), ..., (Bd'(vjp)) ,
are listed in descending order. Denote this new degree sequence by .
Since the subbranch degree sequences of vj are listed in descending order in
My , the new degree sequence M^ satisfies the OSbC. Since Mis a rearrangement
of A with di = d(vj) = dj + 1 > 0, d, = d(vi) — 1 = cfi — 1 for some i > 1, and
A satisfies the DSC, the new degree sequence M^ satisfies the conditions (a) and
(b) of the definition of the DSC. Since mdR(vk) is the perfect sequence of T', it
satisfies the conditions (c) and (d) of the definition of the DSC. Clearly, each section,
(Bd'(vjt)), satisfies the conditions (c) and (d) of the definition of the DSC. Therefore
the new degree sequence satisfies the DSC. Thus the new degree sequence M^
satisfies both the DSC and the OSbC.
For the given nonnegative integer sequence A = (di, 0Í2,..., dn), the set
{Myjd(vj) — d(vi)} is a non-empty finite set, since A 6 {MyJd(vj) — d{iq)}.
Therefore there is a maximum element in this set. Let Pa = max{M^|d(vj) = d(iq)}.
Now A determines a unique tree T. Suppose B is another vertex sequence for T
satisfying the DSC and the OSbC. Then {M®|d(ug) = d(vj)} = {M^|d(u¿) = ¿(iq)},
where vq is an analogue of vj, by Proposition 3.6.5. Therefore Pa is the perfect
sequence for T.
By the above explanation together with Proposition 3.6.2 and Theorem 3.4.12,
we have the following theorem.

50
Theorem 3.6.7 Let A = (di, d2,..., dn) be a nonnegative integer sequence satisfy¬
ing the DSC and the OSbC with d\ > d¿ for i — 2,3, ...,n. Then A = Pa if and
only if A is the perfect sequence of an n-tree.
In the following example, we show the degree sequence B in Example 3.6.6
is the perfect sequence of the tree.
Example 3.6.8 In Example 3.6.6,
A — (di, d2,..., dif) — (3,2,2,1,0,0,2,0,0,1,0,0)
satisfies the DSC and the OSbC and d2 = d3 = d7 = d\ — 1. That is, d(v2) = d(v3) —
d{v7) = d(tq). Thus max{M^|j = 2,3,7} is the perfect sequence of the tree. For
v7 we have that RB(v7) = {v\,v2,... ,v7} U {uio, i>n, tq2} and (Bd'(y3)) = (0) =
(Bd'(v9)). Thus T' = {RB'(y7), E(RB'(v7)} is the subtree produced by the root
subbranch of v7. (See Figure 3-7.)
Since v7 € A(v2), we get the v7-vertex sequence set
SV7(T') = { (u2,Ui,t;io,Uii,Vi2,i;3,V4,t;5,t;6),
(U2, VU U10, vn, Vi2, u3, v6, v4, v5),
(u2, Ui, t>i2, U10, t>n, V3, V4, V5, V6) ,
(u2, Ui, Ü12, U10, vn,v3, vG, v4, vs),
(v2, V3, V4, V5, U6, Ui, U10, Vu, V12) ,
(u2, ^3? ^4? ^5) ^6? ^15 ^12} ^10) ^ll) j
(W2, t>3, Ü6, U4, «5, Vi, 1710, V11, V12) ,
(V2, V3, V6, V4, VS, VU V12, V10, VU) }.
Then the v7-degree sequence set of SV7(T') is
DV7{T) ={ (2,2,1,0,0,2,1,0,0),
(2,2,1,0,0,2,0,1,0),
(2,2,0,1,0,2,1,0,0),
(2,2,0,1,0,2,0,1,0),
(2,2,1,0,0,2,1,0,0),
(2,2,1,0,0,2,0,1,0),
(2,2,0,1,0,2,1,0,0),
(2,2,0,1,0,2,0,1,0) }
= { (2,2,1,0,0,2,1,0,0),
(2,2,1,0,0,2,0,1,0),
(2,2,0,1,0,2,1,0,0),
(2,2,0,1,0,2,0,1,0) }.

51
Therefore we have the maximum sequence mdR(v7) in DV7(T') as mdR(y7) —
(2,2,1,0,0,2,1,0,0). Since mdR(v7) = (2,2,1,0,0,2,1,0,0) > (0) = {Bd'{vs)) =
(fld'M), we have = (3, 2, 2,1,0,0, 2,1,0,0,0,0).
In the same way, we get
M* =(3,2,1,0,0,2,1,0,0,2,0,0)
M* =(3,2,2,1,0,0,2,0,0,1,0,0).
Therefore max{MVj\j = 2,3,7} = MV7. Since
=(3,2,2,1,0,0,2,1,0,0,0,0)
> (3,2,2,1,0,0,2,0,0,1,0,0)
= A,
M¿), which is B in the Example 3.6.6, is the perfect sequence of the tree.
Vll v10 Vi U12
V2
V5 V4 V3 V6
Figure 3-7
By the Tree Classification Theorem (Theorem 3.5.5) and the previous example,
we have the following important corollary.
Corollary 3.6.9 Let A = ( fying the DSC and the OSbC. If d4 — 1 > for every i = 2,3,..., n, then A is the
perfect sequence for an n-tree.
We will see in the following example how we construct a tree shape from a
given maximal degree sequence.
Example 3.6.10 Let D = (ofi, d2, • • •, dw) = (3,2,1,2,0, 0,1,0,0,0) be a maximal
degree sequence. Then we expect a 10-tree, T. Let s = (tq, v2,..., u10) be the vertex
sequence having D as the degree sequence. Then d(v 1) = dx = 3, so the vertex rq

52
has 3 edges. (See Figure 3-8.) Though there is no specific order for these edges, we
label them as ei,e2,e3 for the convenience of explanation.
e2
e3 vi ei
Figure 3-8
From the definition of vertex sequence (Definition 3.3.4) u2 is one of the
elements of the adjacent set of iq, A(iq). Without loss of generality, let ej =
{ui,u2}. Then u2 has 2 ‘new’ edges, e4 and e5, since d2 = 2. Let e4 = {u2,u3}.
By the same reasoning, v3 has 1 ‘new’ edge, named e$. So e& = {u3,u4}. Since
d4 — 2, v4 has 2 ‘new’ edges, named e7 and e8. (See Figure 3-9.)
e2
^5
e8
e3 V\ el l>2 e4 V3 ^6 v4 e7
Figure 3-9
Since d5 = 0 and d6 = 0, neither v5 nor v$ has a ‘new’ edge; that is, u5
and v6 are end vertices. On the other hand, v3 and ve are the adjacent vertices of
t>4, since d4 = 2. Thus v7 is an adjacent vertex of one of {u4, u2, u3}. Since d3 = 1
and v4 is the adjacent vertex of v3, v3 is not an adjacent vertex of v7. Since
d2 = 2 and we have only one adjacent vertex of u2, namely v3, we add an adjacent
vertex, v7. (See Figure 3-10.)
v7
ve
e2
e8
e3 v4 ci V2 e4 V3 e6 v4 e7 v5
Figure 3-10

53
Since d7 = 1, v7 has 1 ‘new’ edge which connects with vs. Since d8 = 0,
there is no‘new’edge from v». Since we covered up to <¿2; the remaining two vertices,
vq and i>io, are adjacent vertices of tq which has two ‘unused’ edges, e2 and e3.
Let e2 = {u^ug}, and e3 = {ui,uio}- Since we are thinking about an unlabeled tree
and we put the edge names for our convenience, there are no specific names for the
vertices and edges. Thus we get the following 10-tree. (See Figure 3-11.)
(«9)
(uio) (t>i)
M
" M
(n2) M
Figure 3-11
M
M
3.7 Extended Partitions and the Algorithm
In Definition 3.5.3 and the following explanation, it was shown that any tree
can be represented by a unique perfect sequence. So, in this section, we develop an
algorithm to produce all the perfect sequences for the n-trees for any positive integer
n. From these perfect sequences, we get the shape of each n-tree as shown in the
previous example, as well as the total number of n-trees.
From Definition 3.4.1 and the First Reduced Degree Sum Formula (Theorem
3.4.6), we see that a degree sequence for an (n + l)-tree always has n + 1 nonnegative
integers as entries and these integers always add up to n which is the number of
the edges of the given (n + l)-tree. Thus, our first step in listing all possible degree
sequences is to form ordered partitions of the integer n as sums of smaller positive
integers.

54
Definition 3.7.1 An ordered partition of the positive integer n is a sequence of
positive integers, (pi,p2, • • • ,Pm), such that Yl?=i Pi = n-
Note that 1 < m < n in the definition above.
Definition 3.7.2 An extended partition of an ordered partition (pi,p2, • • • iPm ) °f
the positive integer n is a sequence of n + 1 nonnegative integers, (eh, d2, • • •> dn+i) 5
such that
(i) di=pi, di = ni and
(ii) there is a subsequence (djx , dj2,..., djm) such that djk = pk for k = 1,..., m.
Note that any entry d¿ in the extended partition must be 0 unless it is a
member of the ordered partition. It is important that different extended partitions
are produced not only by the different ordered partitions but can also be produced
by the same ordered partition.
For example, let (di, d2,-.., d„+i) and (d'l5 dj,..., d|,+1) be ^wo extended
partitions of an ordered partition (pi,p2,... ,pm), and let (dj,, dj2,..., djm) and
«.< djm) be subsequences, respectively, such that djl = di = p\ = d'ti and
djk = pk = d'tk for k = 2,3, ...,m. Assume that ji ^ ti for at least one i. Then
there are sections of zeros with different lengths between and djk, and d't(k
and d[k, respectively. So the extended partitions are different. Thus the order is
very important in an extended partition.
Also note that the length of the extended partition of n must be n + 1. So,
for any given ordered partition, (pi,p2^... ,pm), we need to add n —m + 1 zeros to
make an extended partition.
Proposition 3.7.3 Let (ch, d2,..., dn+i) be an extended partition of an ordered
partition P = (dj,, dj2,..., djm) of the integer n. Then this extended partition sat¬
isfies the First Reduced Degree Sum Formula (Theorem 3-4-6).

55
Proof From the given extended partition (d\, d,2,..., dn+1) of an ordered partition
(dji ,dj2,..., djm) of the integer n, we have that E”=u ^ = £!=i 4 = n an<^
+ £"¿2 (^« ~ 1) = di + £fcl2(4 _ 1) + Ei=i, d, = 4 + £L2(4 -1) + £K wit ~ i)
= EZU 4 + CU-i) + £K w(-i)
= n + (m - l)(-l) + (n-m + 1)(—1),
= n — ra + 1—n + m — 1 = 0. T
Even though we have a very large number of extended partitions of n, some of
them can not be a maximal degree sequence for an (n + l)-tree. So we first eliminate
some ordered partitions that could not possibly result in maximal degree sequences.
To do so, we need to check some properties of perfect sequences.
Proposition 3.7.4 Let D¡ = (di, d2) • • •, 4+i) be the perfect sequence of a given
(n -f l)-íree T. If d{ / 0 and d,+1 = 0, then 0 = dt+i = d,+2 = = d,+d,.
Proof Since d¿ ^ 0, the vertex v¡ has d¿ ‘new’edges; that is, there are d¿ adjacent
vertices of v¿ and they are labeled by d¿ consecutive integers after i. Since D¡ is
a perfect sequence, it satisfies the OSbC. That is, the vertex ut+x is a vertex of the
largest degree among the adjacent vertices of v¿. But c/,+1 = 0. Therefore we have
dj = 0 for j = i + 2, i + 3,..., i + d,-. J_
This proposition gives us a method to make an extended partition from an
ordered partition by adding zeros. That is, we know certain extended partitions
cannot be perfect sequences just by checking this property.
Proposition 3.7.5 Let D-s = (cfi, d2, • • •, 4+i) be the perfect sequence of a given
(n + l)-íree T. If n > 2, then d\ > 2.
Proof If d\ = 1, then dt- = 0 for every i = 2,3,..., n + 1 by Proposition 3.6.2.
Since n > 2, there are at least 3 vertices in T. Thus we have that —1) =
1 + £"=2(—l) ^ l + (—1 — 1) = —1 t^O. This is a contradiction to the First Reduced
Degree Sum Formula. Therefore efi > 2. !_

56
According to the above proposition, the only trees whose perfect sequences
start with 0 or 1 are the tree with zero edge and the tree with one edge. The only
degree sequence representation of a tree with zero edge is (0), and the only degree
sequence representation of a tree with one edge is (1,0). These are easy enough to
visualize and count, so for the algorithm we work on the assumption that we are
listing trees with at least two edges. Then the first term of the perfect sequence will
be at least 2.
The following proposition shows that we have an arc shape tree if the first
term of the perfect sequence is 2.
Proposition 3.7.6 If D-s = (efi, ¿2, • • •, dn+i) is ¿he perfect sequence of an (n + l)-tree
T and d\ — 2, then di = 1 for i = 2,3,..., n — 1 and dn = dn+1 = 0.
Proof By Proposition 3.6.2 we have that € {0,1} for any i = 2,3,..., n + 1,
since di = 2. Then the possible highest degree in T is degree 2. So T is just a
linear tree with n edges. That is, there are exactly two end vertices which cannot be
v\. Therefore there are exactly two zeros in the sequence. Write a degree sequence
for T, in the order of the vertices in the following figure.
tfn+l V1 v2 V3 Un_x Vn
Figure 3-12
That is, start at a vertex adjacent to an end vertex and move in the longer
direction first. Clearly the degree sequence resulting from this method contains
the two zeros for vn and un+i. Then we will get a degree sequence of the form
D = (2,1,1,..., 1,0,0). For the same tree, the numbers in a degree sequence cannot
change, but only the order in which they occur can change by changing the position
of zeros. And, clearly, the new ordered sequence < D. So the sequence D must be
the perfect sequence of T. That is, D¡ = D. ±

57
Most of the extended partitions of n start with a 1 or a 2. Since the last two
propositions classify trees whose degree sequences start with these values, we do not
need to consider ordered partitions that start with 1 or 2.
Proposition 3.7.6 shows that for any given number n of edges there is only one
tree with n edges whose perfect sequence starts with a 2. In particular, it is a linear
tree with n edges, since d,= 1 for i = 2,3,..., n — 1. We add this sequence at the
end of the algorithm.
By Propostion 3.6.2, we do not need the ordered partitions in which the largest
summand appears more than once or is not the first term in the ordered partition.
If an ordered partition has either one or two entries only, then we can produce
only one extended partition for each case. See the following Proposition 3.7.7 and
Corollary 3.7.8.
Proposition 3.7.7 Let D-s — (dx, d2, • • •, dn+i) be the perfect sequence of an (n +1)-
tree T with d2 = 0. Then di — n and d, = 0 for all i > 1.
Proof: Let s be a vertex sequence yielding a perfect sequence D¡ of an (n + 1)-
tree T. Since D-s is a perfect sequence and 0 = d2 = d(v2) — 1, all of the vertices in
the tree other than v\ have degree 1, so all the terms in Dg other than d\ must be
zero. By the First Reduced Degree Sum Formula (Theorem 3.4.6), we have d\ — n. J_
Corollary 3.7.8 If an ordered partition P — (p\,p2) of n has only two entries,
then the produced extended partition, (d1? d2,..., dn+i), from, P is such that dx =
Pi, d2 = p2 and d¿ = 0 for i = 3,4,..., n + 1.
We demonstrated several necessary conditions which apply to a perfect se¬
quence of a tree. Unfortunately, while these conditions are sufficient in almost all
cases, there are certain sequences which, although they satisfy all the conditions, are
not perfect sequences of the trees they represent. The problems usually arise when
the tree has two or more vertices of the largest degree. We saw a counterexample in
Example 3.6.6.

58
Note that in the Example 3.6.6 the two degree sequences A and B have the
same ordered partition, P = (3,2,2,1,2,1). That is, our algorithm produces the
degree sequence B as one of the extended partitions of the ordered partition P.
Therefore all we need to do is just drop the degree sequence A from the result of
our algorithm.
Each step described so far has involved checking each member of a list of
possible degree sequences and removing those which could not be the perfect sequence
of a tree. The last step is no exception. Given a degree sequence, we check if there is
more than one vertex of maximal degree. If there are more than one (in Example 3.6.6,
both vertices iq and vj = u\ have the same degree, 3), we write degree sequences
for the tree starting at each such vertex. If the largest of these with respect to
the ordering < is the sequence we are checking, then it is a perfect sequence. If
some other vertex gives a larger degree sequence, it is false that the sequence we are
checking is a perfect sequence. This is a surefire approach in that, given a degree
sequence, it checks for a perfect sequence by considering every possible alternative
for the maximum and seeing if the sequence in hand is the largest.

CHAPTER 4
TREES AS INVARIANTS FOR BRAIN TUMOR SHAPES
4.1 Introduction
A sphere packing treatment plan for a given brain tumor is produced in the
first chapter. This plan can be represented numerically as a list of parameters of
the dose spheres. In particular, the list consists of the coordinates of the dose center
in millimeters along the lateral (Lat), anterior-posterior (AP), and axial (Ax) axes
together with the radius of the dose sphere in millimeters. From these data, we
can assign a unique corresponding graph by matching a sphere with a vertex and
matching the adjacency of two spheres with an edge. Then, by using the notion of
cutvertex, we give an order to the vertex set. We use this order to decide which edges
to choose in order to obtain a unique maximal tree contained in the graph.
Our main theorem of this section is Theorem 4.5.6: The perfect sequences are
invariants of the shapes of arbitrary brain tumors. We then develop a Mathematica
program (see Appendix B) to get a maximal tree of the graph for a sphere packing
plan for a given brain tumor. In the near future, we plan to use this program to
classify all the treatment plans which have been done at the Brain Institute of the
University of Florida. We hope that the classification gives some insight about brain
tumor shapes.
In this chapter, we assume that a brain tumor is connected, so the graph
representation for any brain tumor is a connected graph.
59

60
4.2 Definitions and Some Properties
Definition 4.2.1 A graph G — (V(G), E(G)) consists of a vertex set V(G) =
{ui,..., vn} and an edge set E(G) = {ei,..., em} C {{u, v}|u, v G V(G)} in which
each edge is an unordered pair of two distinct vertices u and v, and the edge {u, u}
is said to join u and v. For an edge {u,v} we say that u and v are adjacent
vertices and the vertex u and the edge {u,u} are incident with each other, as are
v and {u,t>}. From now on, |P(G')| denotes the number of vertices and \E(G)\
denotes the number of edges. For a vertex u € P(G), the vertex set
Ag{u) = {u € G\v is an adjacent vertex of u}
is called the adjacent set of u in G.
Since there is no loop represented by a single edge, in a graph in this article,
no vertex is adjacent to itself.
Definition 4.2.2 We say that a graph G is isomorphic to a graph H if and only
if there exists a bijection / : V(G) —>■ V(H) such that {u,u} £ E(G) if and only if
{/(«) ,f(v)}€E(H).
Definition 4.2.3 A walk is an alternating sequence (u0, e\, iq, e2,..., e*,, Vk) of ver¬
tices and edges s.t. u,_i / for i — 1,2and e, = for every i. A
cycle is a walk (i>o, ei, iq, e2,..., e*, Vk) with k > 3 in which v0 = Vk is the only
vertex repeated. A path is a walk with no repeated vertex. A (u,v)-path is a path
with the first vertex u and the last vertex v. A (u,u)-path is a vertex u. The length
of the (u,v)-path, l(u,v), is the number of edges in the (u,v)-path. So the length of
(u, u)-path is 0. The distance d(u,v) between two vertices u and v is the length of
a shortest path joining them, if any; otherwise d(u,v) = oo.
Note that a path of length >1 is of linear shape.
Definition 4.2.4 Let G — (V, E) be a graph where V = {vi, u2,..., un}. Then,
for any vertex v € V, we say that PG(v) = j if and only if

61
j = min{f € {1,2,..., n}| there is a (v, u,)-path in G}.
We call Pg(v) the path-value, or simply the P-value of the vertex v in G.
Definition 4.2.5 Let G = (V,E) be a graph. For any nonempth subset V C V,
let F = { {u,v} £ E \ u,v € V } C E. For any subset F' of F, the graph
G' = (V', F') is called a subgraph of G. A graph G is connected if and only if every
pair of vertices are joined by a path. A graph having no cycle is acyclic. Note that an
acyclic graph can be disconnected. A forest is an acyclic graph. A tree is a connected
acyclic graph. An n-tree is a tree with n vertices.
Definition 4.2.6 The degree of a vertex v, denoted by d(v), is the sum of the
number of edges incident with v.
Definition 4.2.7 A cutvertex of a graph is a vertex of degree greater than 2, whose
removal increases the number of components of the graph.
Definition 4.2.8 A nonseparable graph is connected, nontrivial, and has no cutver¬
tex. A block of a graph is a maximal nonseparable subgraph.
Thus if v is a cutvertex of a connected graph G, then G — v is disconnected.
Note that the traditional definition of cutvertex (usually called cutpoint in graph
theory) does not have the degree condition. But we do not want to separate any
linear shape subgraph into several blocks because there is no difficulty in handling
linear shape subgraphs in our algorithm. Thus we do not call a vertex with degree
2, a cutvertex in this chapter, even though it separates the graph.
And if H is a block, then none of the vertices in H is a cutvertex of H.
Note that if there is no cutvertex in G, then the graph G itself is a block. This
definition differs from the definition of block in Harary [5], because we have a different
definition of cutvertex. Because of this, we have Proposition 4.2.10 below.
Proposition 4.2.9 If a graph G is of linear shape, then the maximal tree of G is
G itself.

62
Proposition 4.2.10 [5] Let v be a vertex of a connected graph G. The following
statements are equivalent:
(1) v separates G and d(v) > 2.
(2) There exist vertices u and w distinct from v such that v is on every (u,w)-
path.
(3) There exists a partition of the set of vertices V(G) — {t>} into subsets U and W
such that for any vertices u € U and w € W, the vertex v is on every (u,w)-path.
Definition 4.2.11 Let G = {V, E} be a graph. For any vertex v € V we define
the edge subset Ig(v) C E(G) by Iq{v) — {{v,u} € E(G)\ u € Ag(u)}. We call
Ig(v) the incident edge set of v in G.
Clearly, Ig(v) contains all the incident edges of the vertex v in G.
Definition 4.2.12 Let G — {V,E} be a graph. For any subset F of E we call the
set V(F) = {u € V(G)| there is an edge e 6 F such that v and e are incident }
the F vertex set.
Proposition 4.2.13 [5] The following statements are equivalent for a graph T :
(1) T is a tree.
(2) Every two vertices of T are joined by a unique path.
(3) T is connected and |Vr(T')| = + 1.
(4) T is acyclic and |P(T)| = \E(T)\ + 1.
Note that if a tree has n vertices, then there are exactly n — 1 edges.
Definition 4.2.14 Let G = {V, E} be a connected graph and E' a subset of E.
A connected subgraph T = {V, E'} is called a maximal tree of G if and only if
T' = {V, E' U {e}} contains a cycle for any edge e € E — E'.
Our work in this chapter produces a particular maximal tree for any given
graph. So we have the following (well-known) theorem.
Theorem 4.2.15 Every connected graph has a maximal tree.

63
Example 4.2.16 Let G be a graph with 14 vertices and 18 edges as in the figure
4-1 below.
There are 3 cutvertices, 6, c, e. So the given graph G can be separated into
6 blocks as follows:
Gi = {V1,El} = {{a,b}, {{a,b}}},
G2 = {V2,E2} = { { b,c,k,l,m,n },
{ {b,c},{c,k},{k,l},{l,m},{m,n},{n,b},{b,k},{k,m} } },
G3 = {f/3, £3} = { { c, d, e }, { {c, d}, {d, e} } },
G4 = {V4,E4} = { { ej,g }, { {e, /},{/, g], {g, e} } },
G5 - {V5,E5} = { { e, h,i }, { {e,h},{h,i} } }, and
G6 = {V6,£6} = { { e,j }, { {e,j} } }.

64
4.3 Graphs for Sphere Packing Plans
The sphere packing treatment plan S having n spheres for a given brain tu¬
mor, from Chapter 1, consists of the centers, ci, C2,..., Cn, and the radii, j*i, r2,..., rn,
of the spheres with c, = (x¿, y¿, 2,) for every i, where £,• is for the AP coordinate,
y¡ is for the Lat coordinate and Z{ is for the Ax coordinate of the center, c,-. (Note
that the physicians use diameters, rather than radii, in their data sets.)
We consider the spheres to be the vertices of a graph, and we assign the vertex,
V{, for the sphere centered at c¿ with radius r¿. Now we can assign an edge between
two vertices if the spheres assigned by these two vertices meet. In Chapter 1, we
noted that the collimators do not deliver clear cut boundaries for spheres-there is
some spillover. The neurosurgeons currently assume that the portion between two
spheres gets enough dosage of radiation if the distance between the centers of the
two spheres is less than or equal to eleven tenths of the sum of the radii of the two
spheres. Thus for any two vertices assigned for two spheres satisfying this condition
we assign an edge between them. Then we have the following proposition.
Proposition 4.3.1 Let G — {V, E} be a graph for the sphere packing plan of a
brain tumor. For any vertices and Vj in V, d(c,-,Cj) < y4(rt- + rf) if and only if
£ E. Thus we have the following edge set;
E = {{z, j}\d(ci,Cj) < (ll/10)(r, -f rj) where i < j, i = 1,2,...,n — 1} .
Since any brain tumor is connected in this article, there is at least one (u, v)-
path in G for every two vertices, u and v in G. Thus we have the following propo¬
sition.
Proposition 4.3.2 The graph G = {V, E} produced by the above method for a
given tumor is a connected graph.
Since we are dealing with unlabeled graphs in this article, we have the following
proposition.

65
Proposition 4.3.3 For any sphere packing treatment plan for a given brain tumor,
there exists a unique graph that we assign to the treatment plan.
By means of the above method, we get a unique connected graph G = {V, E}
for the sphere packing treatment plan for the tumor. This graph does not have the
information about the positions of the dose spheres as an aspect of the structure and
looks only at how the spheres are attached. It is clear that the graph G does not
have a loop or multiple edges between two vertices, since every edge is assigned for
two different vertices which are assigned for two attached spheres.
4.4 Order in the Vertex Set
To get a maximal tree from a graph we may need to delete some edges in the
graph. Thus we need to label each vertex in order to choose certain edges to delete,
even though we are dealing with an unlabeled graph in this chapter. So there is no
specific meaning for this labeling except that it is only used to choose a maximal tree
for the graph.
Example 4.4.1 Let P be the sphere packing treatment plan (see Figure 4-3) for a
given brain tumor.
Figure 4-3
Then, by Proposition 4.3.3, there is a unique graph G for the sphere packing plan.
The graph is given below in Figure 4-4.
P
Figure 4-4

66
There are two different isomorphism classes, namely T\ and T2, of maximal trees
for the graph G. That is, we could choose either of these for a maximal tree to assign
the graph G. Since, in our algorithm, we want to get the same maximal tree for a
given graph every time, we need certain rules to get the same tree. (See Figure 4-5.)
Tx T2
Figure 4-5
In this chaper, the graph comes from a sphere packing treatment plan. That is,
the main shape of the brain tumor depends on the size of spheres and the connection
between spheres. Even though we cannot keep the information about the size of
spheres in the graph, it seems to us that the bigger the sphere is, the more it has a
chance to get attached to more spheres. So we assume that the larger degree vertices
play a more important role in classifying the tumor shapes into trees than the smaller
degree vertices. That is, by choosing a largest degree vertex first, the shape of the
tumor is apparently most closely preserved. Thus, in the previous example, we choose
the vertex q or s as the starting vertex; then we pick all the incident edges. Then
we get the tree Tx. Therefore we want to choose the tree Tx for the maximal tree of
G. Note that the tree T2 produces a linear graph which does not show the shape of
the tumor as closely as Tx.
But there are some vertices which are more important than the bigger degree
vertices. Recall the graph G in Example 4.2.16. Then, in the block G2, it is clear
that the vertex k has the largest degree, 5. But the cutvertices, b and c, play a
critical role in obtaining a maximal tree of G. So, to get a maximal tree for a graph
G, we want to start at the cutvertices of G first, instead of the vertices with the
largest degree. So we want to label the cutvertices first. On the other hand, there

67
are different kinds of ‘cutvertices’ in some graphs. The following example shows such
a case.
Example 4.4.2 Let G = {V, E} be the graph in Figure 4-6.
u9
u 7
u 5
Then there are two cutvertices of the graph G, namely ui and Uio. Us¬
ing these two cutvertices, the graph G is separated into four subgraphs, namely
G\, G21 G3 and G4. (See Figure 4-7.)
Then the subgraphs Gi, G2, and G3 are blocks, but the subgraph G4 con¬
tains its own cutvertices, u4 and w8, which are not cutvertices of the graph G. We
separate the subgraph G4 into four subgraphs by using its cutvertices u4 and ug.

68
(See Figure 4-8.) Note that there is no specific order in labeling the subgraphs. So
at this moment, we relabel the subgraphs of G by H\, H2, ..., H7.
u9
Figure 4-8
Then, again, the subgraphs Hi, H2, ..., He are blocks, but the subgraph Hj
contains its own cutvertex, ue, which is neither a cutvertex of the graph G nor a
cut vertex of the subgraph G4. We separate this subgraph H7 into two subgraphs by
using ue. (See Figure 4-9.) And we relabel the subgraphs of G by B\, B2, ..., Bg.
Therefore {«2,^10} is the set of cutvertices of the graph G. But {u4,u8} is
the set of the cutvertices of a subgraph which is produced after the first separation
using the cutvertices of the graph, and ug is the cutvertex of a subgraph which is
produced after the second separation.
For the purpose of labeling the vertices in a graph, we separate these cutvertex
sets for different levels, from each other.
Definition 4.4.3 Let G be the graph assigned for a given sphere packing plan
P. Then the vertex set V(G) of the graph G is the union of two disjoint subsets,
namely Ci(G) and C[(G), where Ci(G) = {v £ V(G)\ v is a cutvertex of G}

69
and C[(G) = V(G) — Gi(G). We call C\(G) the first step cutvertex set of G. For
every i = 2,3,..., |V(G)|, we define a subset G,(G) of Gf_1(G) as the collection
of the vertices of C^fiG), which are cut vertices of a subgraph produced after the
(i — l)-th separation using the elements of G,_i(G). Then we call G,(G) the i-th
step cutvertex set of G, and let C-(G) = Gf_x(G) — G,(G). We assume that there
are finitely many vertices in a given graph. If there exists at least one cutvertex,
then there exists an integer 1 < k < n such that G,(G) ^ 0 for every i < k, and
G¿(G) = 0 for every i > k + 1. Then we call A; the separation step constant of the
graph G, and let GC(G) = (Ct)c(G).
If there is no cutvertex of G then the graph is a block, and we say that the
separation step constant of G is 0.
u9
Figure 4-9.
Proposition 4.4.4 Let |(G)| = n and let the separation step constant of G be
k > 0. Then there is a sequence of integers {mi, m2,..., m*} with 0 < mi < m2 <
...
70
Proof Since k > 0, the first step cutvertex set Gi(G) is not empty. Thus there is
a positive integer, namely mi, such that |Gi(G)| = m\. For every i = 2,3,...,A;,
let m, = m,_i + |G,(G)|. For every i = 2,3,..., k, the t'-th step cutvertex set G¿(G)
is not empty. So we have that m, > m¿_i, and \Ci(G)\ = m¿ — m,_i. J.
Now we label the vertices as follows;
(1) the vertices in Gi(G) are labeled by {1,2,..., mi},
(2) for every i = 2,3,..., k, the vertices in G,(G) are labeled by {m,_i + 1, m¿_i +
2,..., m¡ — 1, m¿}, and
(3) the vertices in GC(G) are labeled by {m*, + 1, m^ + 2,..., n}.
We arbitrarily name the vertices of G,(G) and GC(G), and denote that Gi(G)
}ui, U2,..., }, G,(G) — {^T7i¿_i+i) +21 • • • 5 } for i 2,3,...,/u, and
CC(G) = {umfc+i, umjfc+2,..., un}. It is clear that for every vertex up € G,(G) and
every vertex uq € Cj(G) with i ^ j, we have that p < q if and only if i < j.
And also, for every vertex uv € Ci(G) for every i = 1,2, ...,&, and every vertex
uq € CC(G), we have that p < q. Assume that, for every vertex the center and
the radius of the sphere which is assigned to that vertex u,-, are c¿ = {x¿, z¿) and
r,-, respectively.
We now relabel the elements for each of the sets, Ci(G), (^(G),..., Ck(G)
and GC(G), in the following way:
for every pair of vertices, it,-, Uj, in one of Gi(G), G2(G),..., Gfc(G) and GC(G),
¿ < j if and only if
(i) d(ui) > d(uj)] or
(ii) If d(ui) = d(uj), then r¿ > r¿; or
(ii) If d(ui) = d(uj) and r¿ = r¿, then xt- < Xj\ or
(iii) If d{u{) = d{uj), r¿ = rj, and x,- = Xj, then y¿ < yj; or
(iv) If d(ui) = d(uj), r, = r¿, x, = Xj, and y, = yj, then z,- <
Note that d(u{) represents degree of the vertex it,- in G.

71
It is clear that for any two different spheres, at least one of the coordinates is
different. Thus any two distinct spheres can always be compared by the above order.
Therefore the above order for the vertices of the graph is well defined.
4.5 Maximal Tree from a Graph
In this section, we develop an algorithm that will uniquely select a maximal
tree from the graph which is produced for a given brain tumor. The choice is made
so that the tree will most closely resemble, in our opinion, the shape of the tumor -
that is, of the graph obtained for the tumor.
Let G — (V, E) be the graph for a given sphere packing plan. Let 14(G) =
{ui, u2,..., un} for some positive integer n, where all the vertices in V are labeled
by the method in the previous section.
Define a subgraph T0 — (14(G), F0) with I4(F0) = V = 14(G) and F0 =
0; that is, there is no edge in T0. Then, for every i = 1,2, ...,n, we have that
Pt0(uí) = i. (See Definition 4.2.4.)
(1) For the vertex iq, let Fi = F0 U Ig(ui). (See Definition 4.2.11.) Let =
(14(G), Fi). The subgraph (I4(F1),F1) C T\ is a tree. Thus, if V(Fi) = 14(G), then
T\ — (14(G), Fi) = (14(Fi), F\) is our maximal tree of G. Note that for every vertex
v € 14(Fj) we have that Ptx (u) = 1, and for every vertex Uj V(Fi), we have that
PF (Uj) = j.
(2) Assume that 14(Fi) ^ 14(G). For the vertex U2, there are two cases, that is,
Fji(ii2) = 1 and F7’1(u2) = 2. Now define the edge subset I'G(u2) by /¿(u2) =
IgM - {{u2,u} € E\ Pt,{v) = 1}.
(a) If Ptx(u2) = 1, that is, there is a (ui,u2)-path in the subgraph Ti, then let
F2 = FjU/¿(u2). Each edge in the set {{u2, v} € E\ PTl (v) = 1} = {{u2, v) 6 E¡ve
V(Fi)} makes a cycle with {uuu2} and {tii,u}, since FT,(u2) = 1. Thus there is
no cycle in the subgraph (14(F2),F2), so it is a tree. Let T2 = (14(G), F2) C G.

72
If V(F2) = V(G), then T2 = {V(G),F2) = (V(F2),F2) is our maximal tree of G.
Note that for every vertex v £ V(F2) we have that PtAv) = 1, and for every vertex
uj 0 V(F2) we have that Pj2(uj) — j.
(b) If Pt¡{u2) = 2, then there is no (iti,u2)-path in the subgraph T\.
(i) If there is a non-empty subset {tt2l, u2i,..., u2t} C V(Ig(u2)) where PtAu 2.)
= 1 for i = 1,2,..., t and 2, < 2j for 1 < i < j < t, then let F2 = F\ U PG(u2) U
{{u2,u2l}}. Then the subgraph (V(F2),F2) C G is a tree. Let T2 = (^(G),^) C
G. If V(F2) = I/(G), then T2 = (V(G),F2) = (V(F2), F2) is our maximal tree of
G. Note that for every vertex v 6 V(F2) we have that Pt2 (v) = 1, and for every
vertex Uj V(F2) we have that PtAu])=j.
(ii) If none of the vertices in V{Ig{u2)) has p-value 1, that is, for every vertex
v £ V(Ig{u2)) there is no (i for every vertex w £ F\ and Ptx(v) = 2 for every vertex v £ Ig(u2), the subgraph
(V(F2),F2) is a forest, but not a tree. Let T2 = (V(G),F2) C G. It is clear that
the subgraph T2 cannot be our maximal tree, even if we have that V(F2) — V(G),
because the subgraph T2 is disconnected. Note that for every vertex v £ V{F2) =
V(F\ U IG(u2)) = V{Fi) U V(Ig{u2)) we have that Pt2(v) = 1 if v £ V(F\), and
Pt2(v) = 2 if v £ V(Ig(u2)). And, for every vertex Uj £ V(F2), we have that
ptAuj) = j-
(3) By induction, at the j-th step, the subgraph Tj_i = (V(G),Fj-i) has been
defined. We want to define the subgraph Tj C G. For the vertex ií¿, there are two
cases, that is, Ptj_Au3) — j and Pt}^ {uj) = q for some 1 < q < j. For every
i = 1,2,..., n, define
= min{l £ {1,2,... ,n}\PT]_Aui) = * and ut £ AG{uj) }.
If the set {/ £ {1,2,..., n}|«/ £ V{lQ{uj)) and Pt}-Aui) = * } is empty, then the
vertex (,) is not defined. So there is no edge {tij,umj_,(»)}.

73
(a) Let Ptv_i (uj) = j; that is, for every ¿ = 1,2,..., j — 1, j + 1,..., n, there is
no (uj, u,)-path in the subgraph Tj-i = (V(G),Fj-1). Let
Fj = Fj-1 U (Ut=1{ {uj, umj_i(,)} }) .
Then the subgraph (V(Fj), Fj) is a forest. Let Tj = (V(G),Fj) C G. If V(Fj) =
V(G) and \V(Fj)\ = \Fj\ + 1, then, by Proposition 4.2.13, the subgraph Tj =
(V(G), Fj) = (V(Fj), Fj) is our maximal tree of G.
(b) Let Ptj_, (uj) = q for some 1 < q < j, that is, there is a (u9, tí¿)-path in the
subgraph Tj-i = (V(G), Fj-i). Now let
Fj = Fj-x U (ui=1) {v,j, }) .
Then the subgraph (V(Fj), Fj) is a forest. Let Tj = (V(G),Fj) C G. If V(Fj) =
V{G) and \V(F3)\ — \Fj\ + 1, then by Proposition 4.2.13, the subgraph Tj =
(V(G), Fj) = (V{Fj), Fj) is our maximal tree of G.
Note that if u is an Tth step cutvertex, then d(u) in G is same as d(u) in
Ti_j. Also note that, for every i = 1,2, ...,n, {ui, t¿2,..., tt,-} C ^(F,), that is,
V(G) = V(Fn). We will see in Theorem 4.5.3 that Tn = (V(G),Fn) = (V(Fn), Fn)
is our maximal tree of G. By the Tree Classification Theorem (Theorem 3.6.8), this
tree Tn can be denoted by a unique perfect sequence.
Example 4.5.1 Let G = {V, E) be a graph where V = {ux, u2? • • •, us}? which
are ordered as in the previous subsection, and E = {ei, e2,..., e13} as in the figure
below. Clearly, the graph G does not have a cutvertex, so the order is decided by
only the sizes of the spheres and the coordinates of the centers of the spheres.
^8 ^13 U7 e2 U\
u2 e5 u6
Figure 4-10
U5

74
Let To = (V(G),Fo) be a subgraph of G = (V, E) where Fo = 0, as shown in
Figure 4-11.
u 8
u7
U1
u3.
u4
*U5
• •
u2 U6
Figure 4-11
Since IG(uj) = {ex,e2}, let Fx = F0U/g(ui) = {ex,e2}. Then V^Fx) = {ux,u5,u7}.
Since ^(Fi) = {ui,U5,u7} ^ V{G), the tree (V(Fi),Fi) is not a maximal tree of
G. Let Ti = (V(G),Fi) which is shown in Figure 4-12.
• •
U2 U&
Figure 4-12
We have that mi(¿) = i for i = 1,2,3,4,6, and 8, but mx(5) and mx(7) do not
exist. Since Fr1(u2) = 2 and AG(u2) = {«3, u4, u6}, let
F2 = Fx U (U,=3i4i6{{u2, u,}})
= Fx U Ig(u2)
= {eii e2} U {e3, e4, es} = {ex, e2, e3, e4, es}.
Then V{F2) = {ux,u5,n7, u2,u3,u4,u6} ^ V(G) and the subgraph (V(F2),F2) isa
forest, not a tree. Thus it cannot be a maximal tree of G. Let T2 = (V(G),F2).
(See Figure 4-13.)

75
u2 e5 u6
Figure 4-13
Note that Pt2{ui) = Pt2(u5) = Pt2{u7) = 1, Pt2{u2) = Pt2{u3) = Pt2(u4) =
Pt2{u6) — 2, and Pt2(us) = 8. Thus we have that m2( 1) = l,m2(2) = 2, m2(8) = 8.
Note that m2(i) does not exist for ¿ = 3,4,5,6,7. Since Ui ^ A(u3) and Pt2(u3)—
2, weget F3 = i?2U{{u3, u8}} = {ei, e2, e3, e4, e5, e7}. Since the subgraph (V(F3),F3)
is a forest, but not a tree, it is not a maximal tree of G. Define T3 = (V(G),F3).
(See Figure 4-14.)
u2 e5 u6
Figure 4-14
Note that Pt3(ui) = Pt3(u3) - Ptz{u7) = 1 and Pt3(u2) = Pt3{us) = Pt3(ua) =
Pt3{u6) = Pt3(us) — 2. Thus we have that m3(l) = 5 and m3(2) = 4. Note that
m3(?) does not exist for i = 3,4, ...,8. Since Pt3(u4) = 2, we define F4 - F3 U
{{u4,u5}} = {e1,e2,e3,e4,e5,e7,e8}. Since V(F4) = {uj, v2,..., u8} = V(G) and
|V(F4)| = 8 = |F4| + 1, the subgraph T = (V(F4), F4) is a maximal tree of G. (See
Figure 4-15.)

76
u8 Ur e2 u i
Figure 4-15
At this moment, we have that Pt(u\) = Pt{u2) = ... = Pt(u$) = 1. Then,
for every ut € V(G) with t > 4, we have that Ft = F4. Thus, for every t > 4, the
subgraph (V(Ft),Ft) is same as T = (V(F4), F4).
The next example shows how we get the maximal tree for a given graph
containing some cutvertices.
Example 4.5.2 Let G be the graph in Example 4.2.16. (See Figure 4-1.) Then
Ci(G) = {6,c,e} and Cf(G) — {a,d, f,g,h,i,j,k,l,m,n}. In Ci(G), we have that
d(e) = 5, d(b) = 4 and d(c) = 3. On the other hand, in C'i(G), we have that
d(k) = 5, d(m) = d(n) = 3, d(d) = d(f) = d(g) = d(h) — d(l) — 2, d(a) = d(i) =
d(j) = 1. And C2 = 0- Thus we order the vertices as follows; v\ = e, v2 = b, v3 = c
and v4 = k, {v^v&} = {m,n}, {u7, u8, u9, uio, Un} = {d, f,g, h, /}, {u12,Ui3,ui4} =
{a,i,j}. Assume that v5 = m, u6 = n, u7 = d, v8 = /, u9 = g, ui0 = /i, uu =
l, vi2 = a, i>j3 = ¿, u14 = j, which are decided by the sizes and centers of the
corresponding spheres. (See Figure 4-16.)
Figure 4-16

77
Since IG(vi) = { {ui,t>7},{ui,u8},{ui,M>{ui>uio},{^i,ui4} }, let Fi = IG(vi).
Then V(Fi) = {i7X, v7, v8, v9, u10, i>i4}. Since V(Fi) ^ V(G), the tree 7\ = (V(G), Fi)
is not a maximal tree of G. Thus, for the cut vertex i>i, we have the following sub¬
graph which is T\ = (V(G), Fi). (See Figure 4-17.)
Ü5» • Vn Ui3«
v6» • V4
vu v2 v3
Figure
Since Ptx{v2) - 2, and, for every vertex v € AG(v2) = {u3,u4,u6,^12}, ^ 1,
let F2 = Fi U Ig(v2). The subgraph (V/(F2), F2) is a forest, but not a tree. Thus it
cannot be a maximal tree of G. Now, for the cutvertex w2, we have the following
subgraph which is T2 = (1/(G),F2). (See Figure 4-18.)
4-17
• V\i Ui3«
Since Ag(v3) = {u2,u4,u7}, Pt2{v2) = Fr2(u4) = 2 = Pt2(v3), and Pt2(v7) = 1, let
F3 = F2 U {{u3,u7}}. Then, for the cutvertex u3, we have the following subgraph
which is (1/(F3),F3). Then the subgraph (V(F3),F3) is a tree. But V{F3) -fi V(G).

78
Thus it cannot be a maximal tree of G. We have the following subgraph which is
T3 = (V(G),F3). (See Figure 4-19.)
V5• • Un Ui3«
Since AG{y4) = {u2, u3, u5, i>6, t>n}, and
Pt3{v2) = Pt3(v3) = Pt3(v6) = 1 = Pt3(v4), Pt3{v5) = 5, Pt3(vu) = 11,
we define F4 = F3 U {{^4,^5}} U {{^4,^11}}. Then the subgraph (V(F4), F4) is a
tree. But V(F4) ^ V(G). Thus it cannot be a maximal tree of G. Then, for the
cutvertex v4, we have that the following subgraph which is T4 = (V(G), F4). (See
Figure 4-20.)
For the vertices v5,v6,v7,vs,v9, there is no ‘new’ edge. Thus we get the same sub¬
graph as above, that is, T4 = T5 = T6 = T7 — T8 — T9. But, for the cutver¬
tex U10, we have that U13 £ V(Ig(v 10)) and Pt9{v 13) = 13 ^ 1 = Pt9{vio)- Thus

79
f10 = f9 U{ {^10,^13 } and V(FW) = V(G). Therefore we have the following sub¬
graph which is T = (V(G),Fiq) = (V(Fio), Fio). (See Figure 4-21.)
Clearly, the subgraph T — (F(F’io), Fi0) does not include a cycle. Thus it is a max¬
imal tree of G. That is, for every t — 11,12,13,14, we have that (F(Ft),F¡) =
(V(Fio), F10). Therefore, the subgraph T = (V(Fi0), Fi0) = (V'(G), F10) is the max¬
imal tree of the graph G.
For this resulting tree T = (V(Fio), Fio), choose a vertex sequence s by
s = (vi,v7,v3,v2,v4,v5,vn,ve,vi2,vi0,vi3,vs,v9,vi4). Then the sequence
P = (5,1,1,3,2,0,0,0,0,1,0,0,0,0)
is the degree sequence of s. By Theorem 3.4.12, P satisfies DSC. It is clear that
P also satisfies DSbC. By Corollary 3.6.10, P is the perfect sequence for the tree
T = (V(F10), fio).
Theorem 4.5.3 Let G — {V, F} be a graph for a sphere packing plan where
V = {uj, U2,..., un} with the order described in Section f.f. Let the subgraph
(V(Fn),Fn) of G come from the above procedure. Then V(Fn) = V(G). There¬
fore (V(Fn), F„) is a maximal tree of G and there is a unique perfect n-sequence for
the tree (V(Fn),Fn).
Proof The subgraph (V(Fn), Fn) is connected. In fact, assume that the sub¬
graph (V(Fn),Fn) contains two disjoint subtrees, namely Tx and T2. Then the

80
path-value for every vertex in T\ is different from the path-value for every ver¬
tex in T2. Since the graph G is connected, there is at least one edge e € E(G)
which connects the two subgraphs Ti and T2 and e ^ E(T\ U T2) = Fn. Let
{ {vh,vki},{vj2ivk2}i • • • Avjqivkq} } be the collection of such edges where v]t € T\
and Vk, € T2 for i = 1,2, Without loss of generality, let ji < ji for every
i = 2,3,..., q. Then it is clear that {uj,, } € Fjl by the procedure of getting the
subgraph (V(Fn), Fn). Then £ Fn. This is a contradiction. Therefore the
subgraph (V(Fn), Fn) is connected. The path-value in the procedure of getting the
subgraph (V(Fn),Fn) guarantees that the subgraph (V(Fn),Fn) is acyclic. There¬
fore (V(Fn), Fn) is a tree. It is clear that, for every i = 1,2, ...,n, the vertex set
V(Fi) contains the vertices u\, u2,..., u¿. Thus V(Fn) contains all the vertices of
G, since V{Fn) D {Ul,u2,.. = V{G). That is, V(Fn) = V = V(G). Hence
the subgraph (V(Fn), Fn) is a maximal tree of G. By the Tree Classification The¬
orem (Theorem 3.6.8), it is clear that there is a perfect n-sequence for the tree
(V(Fn),Fn). 1
Definition 4.5.4 The tree shape of a brain tumor is the tree corresponding to that
tumor, or equivalently, its unique corresponding perfect sequence.
Definition 4.5.5 Two tumors have the same shape if and only if their corresponding
graphs obtained from their respective sphere packing plans are the same.
As in Chapter 2, for any brain tumor, we have a unique sphere packing plan
which can be denoted by a graph. From the previous theorem, we have a unique
perfect sequence for the graph. That is, we can assign exactly one perfect sequence to
each brain tumor. So, if two perfect sequences are distinct, then their corresponding
trees, and therefore their respective corresponding graphs and sphere packings, are
also distinct. That is, if two tumors are represented by distinct perfect sequences,
then their corresponding trees are not isomorphic. And their respective graphs and

81
sphere packing plans are not isomorphic. Thus, we may consider their shapes to be
distinct. Therefore we have the following as our main theorem of this section.
Theorem 4.5.6 The perfect sequences are invariants of the shapes of arbitrary brain
tumors.
4.6 Graph from a Maximal Tree
Let T = {V, E'} be a maximal tree for a graph G = {V, E} which is obtained
from the sphere packing plan of a given tumor or solid shape. We observe that we can
recover the original graph, G, from T. Since V(T) = V = V(G) and E(T) — E' C
E = E(G), we need to add the remaining edges to the edge set of the maximal tree to
get the graph G. Since the graph G comes from the sphere packing treatment plan,
each vertex v in G denotes a sphere of the sphere packing plan. Thus, for every
vertex € G, let c, = (a;,-, y,-, zf) and r¿ be the center and the radius of the sphere.
By Proposition 4.3.1, for any two vertices t>, and Vj in V if d(c,-,Cj) < j£(r¿ + r¿),
then the edge is in the edge set, E(G), of the graph G. Thus
E' U {{u¿,i>j} € V x V\ d(ci,Cj) < }¿(r, + r,)} = E.
Therefore we get the graph G = {V, E}.
4.7 Algorithm for Our Maximal Tree
Let G = (V, E) be the graph for a sphere packing plan with |V| = n.
STEP A Find all the ¿-th step cutvertices of the graph G, for 1 < i < n.
STEP B Label the vertices in the vertex set V by the order explained in Section
4.4.
STEP C By induction, for each i = 1,2, ...,n, get the subgraph (1/(F¿),F¿) of
G. Then T = (V(Fn), Fn) is our maximal tree.

APPENDIX A
SHELLING PROGRAM FOR SPHERE PACKING
In this appendix, we provide the C++ program we developed to get the sphere
packing plan for a given tumor shape. The first section shows the program translating
the data from CT scan to the data of appropriate form for the main program, which
is in the second section. Note that this program does not include the score function
which was developed by Thomas Wagner. We refer the reader Appendix E for the
complete article on the sphere packing treatment plan.
A.l Program for Data Reading
// rewrite data file into x,y,z format
// & produce the total # of data, x,y,z min & max
#include
#include
#include
#include
#include
mainO
{ const int nofdata=3370; // comes from data003.txt
int xpositionfnofdata], ypositionfnofdata], zposition[nofdata];
int q;
float a; float b; float c;
int k=0;
{
>
ifstream inClientFile("data003.txt",
while (inClientFile >> a >> b >> c)
{ xpositionfk] = int(floor(a)); //
ypositionfk] = int(floor(b)); //
zpositionfk] = int(floor(c)); //
k++;
>
ios::in);
check the number
check the number
check the number
int xmin=xposition[0];
int xmax=xposition[0];
int ymin=yposition[0];
int ymax=yposition[0];
int zmin=zposition[0];
int zmax=zposition[0];
for (int i=0; i { if ( xpositionfi] < xmin )
xmin = xposition[i];
82

83
>
//
{
>
else
if
( xposition[i]
xmax
=
xposition[i] ;
else
if
( yposition[i]
ymin
=
yposition[i] ;
else
if
( yposition[i]
ymax
=
yposition[i];
else
if
( zposition[i]
zmin
=
zposition[i];
else
if
( zposition[i]
zmax
=
zposition[i];
> xmax )
< ymin )
> ymax )
< zmin )
> zmax )
output the result of centers
ofstream outClientFile("mM003.txt", ios::app);
{ outClientFile << nofdata << setw(10) << xmin << setw(5)
<< xmax << setw(5) << ymin << setw(5) << ymax
<< setw(5) << zmin << setw(5) << zmax << endl;
>
cout << nofdata << setw(5) << xmin << setw(5) << xmax
<< setw(5) << ymin << setw(5) << ymax << setw(5)
<< zmin << setw(5) << zmax;
cin >> q;
>
A.2 Program for Center Searching
// searching the centers for pat 2
#include
#include
#include
«include
#include
int square(int); // function prototype
int MAX(int,int,int); // function prototype
main()
{ const
int
nofdata=3370;
//
check
the
number from mM003.txt
const
int
xmin
=
-32;
//
check
the
number
const
int
xmax
=
-4;
//
check
the
number
const
int
ymin
=
-29;
//
check
the
number
const
int
ymax
=
4;
//
check
the
number
const
int
zmin
=
-57;
//
check
the
number
const
int
zmax
=
-37;
//
check
the
number
const
int
xf ix
=
1-xmin;
const
int
yfix
=
1-ymin;
const
int
zf ix
=
1-zmin;
int xposition[nofdata] , yposition [nofdata], zposition [nofdata] ;

int q;
float a; float b; float c;
int k=0;
84
{ ifstream inClientFile("data003.txt", ios::in);
while (inClientFile >> a >> b >> c)
{ xposition[k] = int(floor(a))+xfix;
yposition[k] = int(floor(b))+yfix;
zposition[k] = int(floor(c))+zfix;
k++;
>
>
const int x = xmax - xmin + 3;
const int y = ymax - ymin + 3;
const int z = zmax - zmin + 3;
const int nc = x*y*z; // number of cells
int cc[nc]; // status value
for (int i=0; i cc[i] = -3;
int xc[nc]
int yc [nc]
int zc [nc]
// x-coord. of c[nc]
// y-coord. of c[nc]
// z-coord. of c[nc]
// setting the x,y,z-coord. for each cell
for (int i=0; i { zc[i] = floor(i/(y*x));
xc[i] = floor((i-zc[i]*y*x)/x);
yc[i] = i-zc [i] *y*x-xc[i] *x;
>
// setting the boundary of tumor by -1
for (int i=0; i { cc[zposition[i]*x*y + xposition[i]*y + yposition[i]] = -1;
// setting the initial of normal by 0
cc[0] = 0;
// setting the normal cell by 0
int h=0;
for (int j=0; j { for (int i=0; i { if ( cc[i] == -3 )
{ if (cc[i-1]==0||cc[i+l]==0I|cc[i-y]==0||cc[i+y]==0)
{ cc[i] = 0;
h++;
>
else;
>
else;
>

>
if (h==0)
break;
else
h=0;
// setting the remaining tumor cell by
for (int i=0; i { if ( cc[i] == -3 )
cc[i] = -2;
else;
}
// checking
// checking
// checking
// checking
// checking
// setting the boundary cells
int maxrad;
int nofmaxrad;
for (int j=0; j<=MAX(x,y,z)/2+l; j++)
{ int k=0;
for (int i=0; i { if (cc[i] == -1 II cc[i] == -2)
{ k++;
if ( cc[i-l] == j I I cc[i+l] ==j || cc[i-y]==j
II cc[i+y]==j II cc[i-y*x]==j II cc[i+y*x] =
{ cc[i] = j+1;
nofmaxrad = k;
}
else;
>
else;
}
maxrad = i;
if (k==0)
break;
else;
// checking the no. of nofmaxrad
// if nofmaxrad > 100, then you have change the number
if (nofmaxrad >100)
{ cout << endl << endl << endl << endl << endl <<
<< endl << endl << endl << endl;
cin >> q;
>
else;
// pick the boxes with maxrad
int suprad[l00];
int TotalNofBox4[l00];
int TotalNofMax=0;
for (int i=0; i { if (cc[i]==maxrad)

{ suprad[TotalNofMax]=i;
TotalNofMax++;
>
else;
cout << TotalNofMax << endl;
cin >> q;
// Count for the number of centers of each case
for (int k=0; k { // reset TotalNofBox4[] to 0
for (int i=0; i<100; i++)
{ TotalNofBox4[i] = 0;
}
TotalNofBox4[maxrad] ++;
int tcc[nc]; // temporary cell notion
for (int i = 0; i tcc[i] = cc[i]; // reassign tcc[i] for 'for' loop
// remove the biggest sphere
for (int p=0; p { int dist = floor(sqrt( square(yc[suprad[k]]-yc[p])
+ square(xc[suprad[k]]-xc[p])
+ square(zc[suprad[k]]-zc[p]) ));
if ( dist < maxrad )
tcc[p] = 0;
else;
}
// setting the remaining tumor cell by -2
for (int i=0; i { if ( tcc[i] >= 1 )
tcc[i] = -2;
else;
>
// setting the boundary cells
for (int j=0; j<=MAX(x,y,z)/2+l; j++)
{ int 1=0;
for (int i=0; i { if (tcc[ij == -2)
{ if ( tcc[i-l] == j || tcc[i+l] ==j || tcc[i-y]==j
II tcc[i+y]==j || tcc[i-y*x]==j || tcc [i+y*x]==j
{ tcc[i] = j + 1;
++1;
>
else;
>
else;
>
if (1==0)
break;

87
else;
>
// check if there is more maxrad pixcel
for (int i=0; i { if ( tcc[i] == maxrad )
{ cout << endl << endl << endl <<
" WARNING!!! WE HAVE MORE 'MAXRAD' PIXCEL AT ("
« xc[i] « « yc[i] « « zc[i] « ")"
<< endl;
cin >> q;
break;
>
else;
>
//remove the smaller or = sphere
for (int j=maxrad-l; j>1; —j)
{ for (int i=0; i { if ( tcc[i] ==j )
{ TotalNofBox4[j]++;
// remove the sphere
for (int 1=0; l { int dist = floor(sqrt( square(ye[i]-yc[1])
+ square (xc [i]-xc [1] )
+ square (zc [i] -zc [1] ) ));
if ( dist < j )
tcc[l] = 0;
else;
>
// setting the remaining tumor cell by -2
for (int i=0; i { if ( tcc[i] >= 1 )
tcc[i] = -2;
else;
>
// setting the boundary cells
for (int j=0; j<=MAX(x,y,z)/2+l; j++)
{ int 1=0;
for (int i=0; i { if (tcc[ij == -2)
{ if ( tcc[i—1] == j || tcc[i+l] ==j ||
tcc[i-y]==j II tcc[i+y]==j I I
tcc[i-y*x]==j || tcc[i+y*x]==j )
{ tcc[i] = j + 1;
++1;
>
else;
>
else;
}
if (1==0)
break;

88
else;
>
>
else;
>
>
// output the result of centers
{ ofstream outClientFile("pat003-NofCenters.txt", ios::app);
{ outClientFile << endl << endl << endl << endl << endl <<
(k+1) << setw(5) << "(" << (xc[suprad[k]]-xfix) <<
<< (yc[suprad[k]]-yfix) << << (zc[suprad[k]]-zfix)
<< ") with radius = " << maxrad << endl;
>
>
for (int j=maxrad; j>1; j —)
{ ofstream outClientFile("pat003-NofCenters.txt", ios::app);
{ outClientFile << setw(4) << j << setw(4)
<< TotalNofBox4[j] << endl;
>
}
>
// Search for centers
for (int k=0; k { int tcc[nc]; // temporary cell notion
for (int i = 0; i tcc[i] = cc[i]; // reassign tcc[i] for 'for' loop
// output the result of centers
{ ofstream outClientFile("pat003-centers.txt", ios::app);
{ outClientFile << (k+1) << endl << setw(5) << "x"
<< setw(5) << "y" << setw(5) << "z" << setw(10)
<< "radius" << endl << setw(5) << (xc[suprad[k]]-xfix)
<< setw(5) << (yc[suprad[k]]-yfix) << setw(5)
<< (zc[suprad[k]]-zfix) << setw(10) << maxrad << endl;
>
}
// remove the biggest sphere
for (int p=0; p { int dist = floor(sqrt( square(yc[suprad[k]]-yc[p])
+ square (xc [suprad [k]]-xc[p])
+ square (zc [suprad [k]]-zc[p]) ));
if ( dist < maxrad )
tcc[p] = 0;
else;
>
// setting the remaining tumor cell by -2
for (int i=0; i -C if ( tcc[i] >= 1 )
tcc[i] = -2;

89
else;
}
// setting the boundary cells
for (int j=0; j<=MAX(x,y,z)/2+l; j++)
{ int 1=0;
for (int i=0; i { if (tcc[ij == -2)
{ if ( tcc[i-l] == j I I tcc[i+l] ==j II tcc[i-y] ==j
II tcc[i+y]==j II tcc[i-y*x]==j II tcc[i+y*x]==j )
{ tcc [i] = j+1;
++1;
}
else;
>
else;
}
if (1—0)
break;
else;
}
// check if there is more maxrad pixcel
for (int i=0; i { if ( tcc[i] == maxrad )
{ cout << endl << endl << endl <<
" WARNING!!! WE HAVE MORE ’MAXRAD’ PIXCEL AT ("
<< xc[i] << << yc[i] << << zc[i] << ")" << endl;
cin >> q;
break;
}
else;
>
//remove the smaller or = sphere
for (int j=maxrad-l; j>1; —j)
{ for (int i=0; i { if ( tcc[i] ==j )
{ TotalNofBox4[j]++;
// output the result of centers
{ ofstream outClientFile("pat003-centers.txt",
ios::app);
{ outClientFile << setw(5) << (xc[i]-xfix)
<< setw(5) <<(yc[i]-yfix) << setw(5)
<< (zc[i]-zfix) << setw(10) << j << endl;
}
>
// remove the sphere
for (int 1=0; l { int dist = floor(sqrt( square(ye[i]-yc [1])
+ square (xc[i]-xc[l])
+ square(zc[i]-zc[l] ) ));
if ( dist < j )
tcc[l] = 0;

90
else;
>
// setting the remaining tumor cell by -2
for (int i=0; i { if ( tcc[i] >= 1 )
tcc[i] = -2;
else;
>
// setting the boundary cells
for (int j=0; j<=MAX(x,y,z)/2+l; j++)
{ int 1=0;
for (int i=0; i { if (tcc[i] == -2)
{ if ( tcc[i-l] == j I I tcc[i+l] ==j II
tcc[i-y]==j II tcc[i+y]==j II
tcc [i-y*x] ==j || tcc[i+y*x]==j )
{ tcc [i] = j+1;
++1;
>
else;
}
else;
}
if (1—0)
break;
else;
}
}
else;
>
>
{ ofstream outClientFile("pat003-centers.txt", ios::app);
{ outClientFile << endl;
}
>
}
>
int square(int a)
{ return a*a;
>
int MAX(int a, int b, int c)
{ int p=0;
p = a;
if (a<=b)
p = b;
else if (a<=c)
p = c;
else;
return p;
>

APPENDIX B
PROGRAM FOR THE MAXIMAL TREE OF A GRAPH
In this appendix, we provide the Mathematica program we developed in joint
work with Matthew Harvey. The program below was written by Matthew Harvey to
get the maximal tree for the graph which we assigned for a brain tumor. Note that
this program does not use the method of ordering for the vertices as in Chapter 3.
Instead this program uses some commands which are in the function package in the
Mathematica program library. If anyone wants to develop a program like this using a
different program language, we recommand the vertex ordering algorithm in Chapter
3.
<< DiscreteMath'Combinatorica'
Off[General::spell]
LexPrecedes[a_, b_] := Module[{i},
For[i = 1, i <= Min[Length[a], Length[b]], i++,
If[ a[[i]] ! = b[[i]],
If[ a[[i]] > b [ [i]] , Return[True] , Return [False] ] ]
3;
If[ Length [b] > Length [a], Return [False] ];
Return[True]]
DegreePoint[a_, k_] := Count[Flatten[ToUnorderedPairs[a]], k];
IndSubgraph[Graph[g_, v_], s_List] := Module[{i, j, newmat, sc},
newmat = g;
sc = Complement[Range[Length[v]], s] ;
For[i =1, i <= Length[g], i++,
For [j = 1, j <= Length [g], j++,
If[MemberQ[sc, i] || MemberQ[sc, j], newmat[[i, j]] = 0]]];
Graph[newmat, v]];
CutPoints[gr_] :=
Select[ArticulationVertices[gr], DegreePoint[gr, #] > 2 &]
NonCutPartitions [gr_] := Module[{partitions, cpts, bees, skeleton},
partitions = {};
bees = Select[BiconnectedComponents[gr], Length[#] > 2 &];
partitions = Union[partitions, bees] ;
skeleton =
IndSubgraph[gr,
Union[
Complement[ Range[V[gr]], Flatten[bees] ],
91

CutPoints[gr]
]
];
partitions =
Union[partitions,
Select[ConnectedComponents[
FromUnorderedPairs[
Select[ToUnorderedPairs[
skeleton], # !=
Intersection[#,
Flatten[
Select[BiconnectedCoraponents[skeleton],
Length [#] >2 &]
]
]
&]
]
], Length[#] > 1
&]
];
Table[
Select[
ToUnorderedPairs[gr] ,
# == Intersection[ #, partitions [[i]] ]
&], {i, Length[partitions]}]]

APPENDIX C
PROGRAM FOR CLASSIFYING TREES BY SEQUENCES
In the first section, we provide the Mathematica program we developed in
joint work with Matthew Harvey. This program was written by Matthew Harvey to
get the maximal degree sequences for any given integer n > 2. And we show the
program results up to n = 10 in the second section.
C.l Program
Off[General::spell]
<< DiscreteMath‘Combinatorica*
DoPartitions[n_] :=
Module[{i, k, templist, returnlist},
If[n == 0, {{}},
returnlist = {};
For[k = n, k >= 1, k—,
templist = DoPartitions[n - k] ;
For[i = 1, i <= Dimensions[templist] [[l]], i++,
returnlist = Append [returnlist,
Flatten[{k, templist [[i]] }]
]
]
];
returnlist
]
];
Filter[thelist_] :=
Select[thelist,
! ( MemberQ[{1, 2}, #1[[1]]] || (Max[#l] != First[#1])
II (Count [#1, Max[#l]] > 1)
)
&]
AddZerosToSeq[part_] :=
Module[{nZeros, n, i, j, k, sum, positions, seqs, withzeros},
If [Length [part] == 1, Return [{Table [If [i == 1, part[[l]], 0] ,
{i, part [[l]] + 1>]>]
];
seqs = {>;
sum = Sum[part[[i]], {i, 1, Length[part]>];
nZeros = (sum + 1) - Length[part];
n = Table[k, {k, 3, sum}];
positions = KSubsets[n, nZeros - 1];
93

Ford = 1, i <= Length [positions] , i++,
k = 2;
withzeros = {First[part]};
For[j = 2, j <= sum, j++,
If[MemberQ[positions[[i]], j] ,
withzeros = Append[withzeros, 0],
withzeros = Append[withzeros, part[[k]]];
k++
]
];
withzeros = Append[withzeros, 0];
seqs = Append[seqs, withzeros]
];
seqs
];
AddZeros[seq_] :=
Flatten[Table[ AddZerosToSeq[seq[[i]]] , {i, 1, Length[seq]}],1]
SubSeqs[dseq_] :=
Module[ {branchlist, branch, workseq, branches, i},
branches = dseq[[l]];
workseq = Drop[dseq, 1];
branchlist = {};
For[i = 1, i <= branches, i++,
dsum = workseq[[1]];
workseq = Drop[workseq, 1];
branch = {dsum};
While[dsum != 0, dsum += (workseq[[l]] - 1);
branch = Append[ branch, workseq[[l]] ];
workseq = Drop[workseq, 1]
] f
branchlist = Append[branchlist, branch]
^ )
branchlist
]
LexPrecedes[a_, b_] :=
Module[{i},
For[i = 1, i <= Min[Length[a], Length[b]], i++,
If[a[[i]] ! = b[[i]],
If [ a[[i]] > b[[i]] , Return [True] , Return [False]
]
]
];
If[Length[b] > Length[a], Return [False] ] ;
Return[True]
]
WeightsQ[dseq_] :=
Module[{i, subseqs, flag},
subseqs = SubSeqs[dseq];
flag = 1;
If[OrderedQ[subseqs, LexPrecedes],

95
For[i = 1, i <= Length[subseqs], i++,
If[ ! WeightsQ[subseqs[[i]]], flag=0, flag*=l ]
],
Return[False]
];
If[ flag == 1, Return[True] , Return[False] ]
]
DegreeSumQ[dseq_] :=
Module [{dsum},
dsum = dseq[[l]];
For[i = 2, i <= Length[dseq], i++,
dsum += dseq[[i]] - 1;
If[dsum == 0,
If [i == Length[dseq].Return[True],Return[False]
]
]
];
Return[False]
]
ExtractSubtree[dseq_] :=
Module [{rtotal},
rtotal = dseq[[l]];
i = 1;
While[rtotal > 0,i++;
rtotal += dseq[[i]] - 1;
];
Take[dseq, i]
]
RewriteTree [dseq_, i_] : =
Module[{branchesoff, subtree, brlength, seqlst},
If[i == 1, Return[dseq]];
branchesoff = i - 1;
While[ ( branchesoff > 1 && Sum[dseq[[k]] - 1,
{k, branchesoff, i - 1}] < 0 ) ||
dseq[[branchesoff]] == 0, branchesoff— ];
brlength = Length[ ExtractSubtree[Drop[dseq, i - 1]] ];
subtree = Drop[dseq, {i, i + brlength - 1}];
subtree[[branchesoff]]—;
seqlst = SubSeqs[Take[dseq, {i, i + brlength - 1}]];
seqlst = Append[seqlst, RewriteTree[subtree,branchesoff]];
seqlst = Sort[seqlst, LexPrecedes[#1, #2] &];
Flatten[{dseq[[i]] + 1, seqlst}]
RightOrderQ[dseq_] :=
Module[{pos, seqs, j, rewritten},
pos = Flatten[ Position[dseq, dseq[[l]] - 1] ];
If[pos == {}, Return[True]];
seqs = {};
For [j = 1, j <= Length [pos] , j++,
rewritten = RewriteTree[ dseq, pos[[j]] ];

96
If[LexPrecedes[rewritten, dseq] && rewritten != dseq,
Return[False]
]
];
Return[True]
LemmasQ[dseq_] :=
DegreeSumQ[dseq] && WeightsQ [dseq] && RightOrderQ[dseq]
CheckLemmas[lstseq_] :=
Module[{retseq, twoseq, i, j, n, p},
retseq = {};
j = 0;
P = 10;
n = Length[lstseq];
For[i = 1, i <= n, i++,
If[ j > n/10,
Print [p, "'/,"];
p += 10;
j = 0
];
j++;
If [LemmasQ [ lstseq[[i]] ],
retseq = Append[ retseq, lstseq[[i]] ]
]
];
twoseq = {2};
If [Length[lstseq] > 0,
For[i = 2, i < Length[ retseq[[l]] ] - 1, i++,
twoseq = Append[twoseq, 1]
]
];
twoseq = Join[twoseq, {0, 0}];
retseq = Append[retseq, twoseq];
retseq
]
DSTRecursive[dseq_, bvert_, hvert_ , trparam_ ] :=
Module[{i, v, tree, tempseq},
v = hvert;
tree = trparam;
If[dseq != {0}, tempseq = Drop[dseq, 1];
For[i = 1, i <= dseq[[l]], i++,
v++;
tree = Append[tree, {bvert , v}];
subseq = ExtractSubtree[tempseq];
tempseq = Drop[tempseq, Length[subseq]];
{tree, v} = DSTRecursive[subseq, v, v, tree];
];
{tree, v}

97
DegreeSeqToTree[dseq_] :=
First[ DSTRecursive[dseq, 1, 1, {}] ] ;
ToGraphics[ckd_] :=
For[t = 1, t <= Length[ckd], t++,
Print [ckd [ [t]]];
ShowGraph[RadialEmbedding[FromUnorderedPairs
[ DegreeSeqToTree[ckd[[t]]] ]
]
]
]
C.2 Some Program Results
The notation for a degree sequence is (d\, d-i,..., dn). But the Mathematica
program produces the sequence notation with { and }. So in this section, any set
notation is actually denotes the degree sequence. Note that an n-tree has n vertices,
so there are n-1 edges.
n=3 total : 1
{2, 0, 0}
n=4 total : 2
-C3, 0, 0, 0}, {2, 1, 0, 0>
n=5 total : 3
{4, 0, 0, 0, 0>, {3, 1, 0, 0, 0},
{2
1
1, 0, 0}

98
n=6 total : 6
{5, 0, 0, 0, 0, 0}, {4, 1, 0, 0, 0, 0>, {3, 2, 0, 0, 0, 0},
{3, 1, 0, 1, 0, 0>, {3, 1, 1, 0, 0, 0}, {2, 1, 1, 1, 0, 0}
n=7 total : 11
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-end of data-

APPENDIX D
LABELED GRAPHS AND TREES
D.l Number of Labeled Graphs and Labeled Trees
To obtain the number of labeled graphs with n vertices, we need only observe
that each of the (”) possible edges is either present or absent. Thus we have the
following theorem:
Theorem D.1.1 [5]
The number of graphs on n labeled vertices is
In Lovász [6], the Priifer-code is used to denote a labeled tree. And, for any
given Priifer-code, one can construct the tree which is denoted by that code. Then,
using the properties of the Priifer-code, Cayley proved the following theorem:
Theorem D.l.2 (Cayley) [8] The number of trees on n labeled vertices is nn~2.
D.2 Sketch of the Construction of a Priifer-code
Among the end vertices of a given labeled tree, pick the one whose label is
minimal. Cut it off from the tree and write down the label of its adjacent vertex.
Repeat this process until only one edge remain. At this point we have a sequence of
n — 1 numbers written down. We call this resulting sequence the Priifer-code of the
tree.
The following example shows how we construct the Priifer-code of a given
tree.
Example D.2.1 Let T be the tree shown below.
3 6
2
1
5
7
8
4
Figure D-l.
There are 5 end vertices, 2, 3,4,6,8. We cut off the smallest labeled end vertex
which is 2. Then the sequence is (1) and the remaining tree is Figure D-2 below.
Then we cut off the next smallest labeled end vertex which is 3, and add the
label, 1, of the adjacent vertex of 3 to the sequence. That is, the sequence is (1,1).
From the remaining tree we cut off the next smallest labeled end vertex which is 4.
Then the sequence is (1,1,1). And we get the tree shape in Figure D-3.
102

103
3 6
15 7 8
4
Figure D-2.
6
15 7 8
Figure D-3.
Among the 3 end vertices, 1 is the smallest labeled one. So we cut off the
vertex 1 and write down the label, 5, of the adjacent vertex of 1. The sequence then
will be (1,1,1,5) and the remaining tree is Figure D-4.
6
5 7 8
Figure D-4.
The remaining tree has two end vertices, that is, it is a linear shape. We cut
off the smaller labeled vertex, 6. And the sequence will be (1,1,1,5,5) . After we cut
off the next smaller labeled vertex which is 5, and get the sequence (1,1,1,5,5, 7) ,
the remaining tree has only two vertices. It’s clear we have to cut off the smaller
labeled vertex which is 7 and the adjacent vertex will be 8. Therefore we do not need
to add this number to the sequence. Hence the last sequence, (1,1,1,5,5, 7), is the
Priifer-code of the given tree T.
If we assign different labels for the tree above, then we will get a different
Priifer-code. But the code must consist of 6 terms which is two less than the total
number of vertices in the tree. And each term can be any number among 1,2,..., 8.
Thus we have the following theorem.
D.3 Reconstruction of a Tree from a Prüfer-code
Let (al5 02,..., a„-2) be a Priifer-code. That is, we expect an n-tree. Let
6i, ¿>2, • • •, 5n_i be the indices of the removed points. Then, for any i, 6, is obviously
different from 61,62,... ,6,_i,a,-. Also, 6,- ^ aj if j > i, because 6, is removed in
the i-th step, so it can not be an adjacent vertex later.
Conversely, if k £ {61? 62,..., 6,_x, a,-, a,+1,..., an_2}, then Vk is an end ver¬
tex of V(T) — {ve»,, v&2, • • •, i’fe.—i}- Thus

104
6, = min{A:|A: £ {61, b2,..., 6,-1, a,, ai+i,..., a„_2}.
Let the sequence (1,1,1,5,5,7) be a Priifer-code. That is, a\ = l,a2 =
l,a3 = l,a4 = 5,a5 — 5, o6 = 7. Then b\ — min{A;|A: ^ {ai, a2,..., a6}} =
min{fc|A; 0 {1,5,7}} = min{2,3,4,6,8} = 2. That is, there is an edge between ax =
1 and b\ = 2. Then 62 = min{A;|A; ^ {6i, a2,..., a6}} = min{A;|A; ^ {2,1,5,7}} =
min{3,4,6,8} = 3. Thus the tree T has an edge between a2 = 1 and 62 = 3. In a
similar manner, we have the following:
63 = min{A:|fc ^ {6i,62,a3,... ,a6}}
= min{fc|A; ^ {2,3,1,5, 7}} = min{4, 6,8} = 4,
(Thus T has an edge {«3,63} = {1,4})
64 = min{fc|fc ^ {61,62,53,a4,a5,a6}}
= min{A:|A: ^ {2,3,4,5,7}} = min{l,6,8} = 1,
(Thus T has an edge {a4,64} = {5,1})
65 = min{A:|A: ^ {61,62,63,64, a5,a6}}
= min{A:|A; 0 {2,3,4,1,5, 7}} = min{6,8} = 6,
(Thus T has an edge {05,65} = {5,6})
b6 = min{A;|A: ^ {6l5 62,63,64,65,05}}
= min{fc|fc ^ {2,3,4,6, 7}} = min{5,8} = 5,
(Thus T has an edge {a6,66} = {7,5}).
And, for the vertex a& = 7, the largest labeled vertex u8 will be the remaining
adjacent vertex, that is, there is an edge {7,8}.
So the tree T has the vertex set V = {1,2,3,4,5,6, 7,8} and the edge set
E = {{1,2}, {1,3}, {1,4}, {1,5}, {5,6}, {5, 7}, {7,8}}, which is the tree shape in the
example above. That is, we can construct a labeled tree from a Priifer-code. Clearly,
different Priifer-codes produce different labeled trees.

BIOGRAPHICAL SKETCH
Tae-Il Yi was born December 16, 1962 in Seoul, Korea. He attended the
Dankook University from 1981 to 1988, earning a Bachelor of Science degree in math¬
ematics education in February 1988. During college, he was in the service of the
Korean Army for 27 months. After college, he entered graduate school at Dankook
University in March 1988, and completed a Master of Science degree in mathemat¬
ics in Feburary 1990. During that time he worked at Yongmoon High School as a
mathematics teacher.
He married Moon-Sil Kim on June 27, 1987. Their first son, Han-Yong
(David), was born April 12, 1988, and their second son, Chang-Yong (Peter), was
born May 3, 1994.
In August 1991 Tae-Il Yi entered graduate school in the Mathematics De¬
partment at the University of Illinois at Urbana/Champaign. After receiving the
degree of Master of Science in August 1994, he moved to graduate school in the In¬
struction and Curriculum Department of the College of Education at the University
of Florida, and in August 1995 transfered to the University of Florida, Mathematics
Department. He pursued a concurent degree in the College of Education and received
a degree of Master of Education in December 1997. During his doctoral work in the
Mathematics Department, he served as a graduate teaching assistant.
He received a Russell Corporation Endowed Dissertation Scholarship Spring
2000. He is a member of the Phi Kappa Phi honor society.
105

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
^ ^
Beverly LHsrechner, Chairman
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
5hilip Boyland
Associate Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is full/- adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy»
^sling
lessor of Mathematic
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
WU
Li-Chien Shen
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Mary G. Kanxowskr
Professor of Teaching and Learning
This dissertation was submitted to the Graduate Faculty of the Department of
Mathematics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.
August 2000
Dean, Graduate School




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