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Chapter 11

Trees

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Problem 1

Which of these graphs are trees?
A.(GRAPH NOT COPY)
B.(GRAPH NOT COPY)
C.(GRAPH NOT COPY)
D.(GRAPH NOT COPY)
E.(GRAPH NOT COPY)
F.(GRAPH NOT COPY)

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Problem 2

Which of these graphs are trees?
A.(GRAPH NOT COPY)
B.(GRAPH NOT COPY)
C.(GRAPH NOT COPY)
D.(GRAPH NOT COPY)
E.(GRAPH NOT COPY)
F.(GRAPH NOT COPY)

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Problem 3

Answer these questions about the rooted tree illustrated.
(GRAPH NOT COPY)
a) Which vertex is the root?
b) Which vertices are internal?
c) Which vertices are leaves?
d) Which vertices are children of $j$ ?
e) Which vertex is the parent of $h ?$
f) Which vertices are siblings of $o ?$
g) Which vertices are ancestors of $m ?$
h) Which vertices are descendants of $b ?$

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Problem 4

Answer the same questions as listed in Exercise 3 for the rooted tree illustrated.
(GRAPH NOT COPY)

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Problem 5

Is the rooted tree in Exercise 3 a full $m$ -ary tree for some positive integer $m ?$

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Problem 6

Is the rooted tree in Exercise 4 a full $m$ -ary tree for some positive integer $m ?$

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Problem 7

What is the level of each vertex of the rooted tree in Exercise 3$?$

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Problem 8

What is the level of each vertex of the rooted tree in Exercise 4$?$

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Problem 9

Draw the subtree of the tree in Exercise 3 that is rooted at
$$\begin{array}{llll}{\text { a) } a} & {\text { b) } c} & {\text { c) } e}\end{array}$$

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Problem 10

Draw the subtree of the tree in Exercise 4 that is rooted at
$$\begin{array}{llll}{\text { a) } a} & {\text { b) } c} & {\text { c) } e}\end{array}$$

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Problem 11

a) How many nonisomorphic unrooted trees are there with three vertices?
b) How many nonisomorphic rooted trees are there with three vertices (using isomorphism for directed graphs)?

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Problem 12

a) How many nonisomorphic unrooted trees are there with four vertices?
b) How many nonisomorphic rooted trees are there with four vertices (using isomorphism for directed graphs)?

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Problem 13

a) How many nonisomorphic unrooted trees are there with five vertices?
b) How many nonisomorphic rooted trees are there with five vertices (using isomorphism for directed graphs)?

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Problem 14

Show that a simple graph is a tree if and only if it is connected but the deletion of any of its edges produces a graph that is not connected.

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Problem 15

Let $G$ be a simple graph with $n$ vertices. Show that
a) $G$ is a tree if and only if it is connected and has $n-1$ edges.
b) $G$ is a tree if and only if $G$ has no simple circuits and has $n-1$ edges. $[\text { Hint: To show that } G \text { is connected }$ if it has no simple circuits and $n-1$ edges, show that $G$ cannot have more than one connected component. $]$

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Problem 16

Which complete bipartite graphs $K_{m, n},$ where $m$ and $n$ are positive integers, are trees?

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Problem 17

How many edges does a tree with $10,000$ vertices have?

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Problem 18

How many vertices does a full 5 -ary tree with 100 internal vertices have?

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Problem 19

How many edges does a full binary tree with 1000 internal vertices have?

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Problem 20

How many leaves does a full 3 -ary tree with 100 vertices have?

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Problem 21

Suppose 1000 people enter a chess tournament. Use a rooted tree model of the tournament to determine how many games must be played to determine a champion, if a player is eliminated after one loss and games are played until only one entrant has not lost. (Assume there are no ties.)

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Problem 22

A chain letter starts when a person sends a letter to five others. Each person who receives the letter either sends it to five other people who have never received it or does not send it to anyone. Suppose that $10,000$ people send out the letter before the chain ends and that no one receives more than one letter. How many people receive the letter, and how many do not send it out?

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Problem 23

A chain letter starts with a person sending a letter out to 10 others. Each person is asked to send the letter out to 10 others, and each letter contains a list of the previous six people in the chain. Unless there are fewer than six names in the list, each person sends one dollar to the first person in this list, removes the name of this person from the list, moves up each of the other five names one position, and inserts his or her name at the end of this list. If no person breaks the chain and no one receives more than one letter, how much money will a person in the chain ultimately receive?

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Problem 24

Either draw a full $m$ -ary tree with 76 leaves and height $3,$ where $m$ is a positive integer, or show that no such tree exists.

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Problem 25

Either draw a full $m$ -ary tree with 84 leaves and height 3 , where $m$ is a positive integer, or show that no such tree exists.

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Problem 26

A full $m$ -ary tree $T$ has 81 leaves and height $4 .$
a) Give the upper and lower bounds for $m .$
b) What is $m$ if $T$ is also balanced?

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Problem 27

A complete $m$ -ary tree is a full $m$ -ary tree in which every leaf is at the same level.
Construct a complete binary tree of height 4 and a complete 3 -ary tree of height 3 .

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Problem 28

A complete $m$ -ary tree is a full $m$ -ary tree in which every leaf is at the same level.
How many vertices and how many leaves does a complete $m$ -ary tree of height $h$ have?

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Problem 29

Prove
a) part (ii) of Theorem 4 .
b) part (iii) of Theorem 4.

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Problem 30

Show that a full $m$ -ary balanced tree of height $h$ has more than $m^{h-1}$ leaves.

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Problem 31

How many edges are there in a forest of $t$ trees containing a total of $n$ vertices?

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Problem 32

Explain how a tree can be used to represent the table of contents of a book organized into chapters, where each chapter is organized into sections, and each section is organized into subsections.

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Problem 33

How many different isomers do these saturated hydro- carbons have?
$$\begin{array}{llll}{\text { a) } \mathrm{C}_{3} \mathrm{H}_{8}} & {\text { b) } \mathrm{C}_{5} \mathrm{H}_{12}} & {\text { c) } \mathrm{C}_{6} \mathrm{H}_{14}}\end{array}$$

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Problem 34

What does each of these represent in a tree that represents the structure of an organization?
a) the parent of a vertex
b) a child of a vertex
c) a sibling of a vertex
d) a sibling of a vertex
e) the ancestors of a vertex
f) the level of a vertex
g the height of the tree

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Problem 35

Answer the same questions as those given in Exercise 34 for a rooted tree representing a computer file system.

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Problem 36

a) Draw the complete binary tree with 15 vertices that represents a tree-connected network of 15 processors.
b) Show how 16 numbers can be added using the 15 processors in part (a) using four steps.

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Problem 37

Let $n$ be a power of $2 .$ Show that $n$ numbers can be added in $\log n$ steps using a tree-connected network of $n-1$ processors.

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Problem 38

A labeled tree is a tree where each vertex is assigned a label. Two labeled trees are considered isomorphic when there is an isomorphism between them that preserves the labels of vertices. How many nonisomorphic trees are there with three vertices labeled with different integers from the set $\{1,2,3\} ?$ How many nonisomorphic trees are there with four vertices labeled with different integers from the set $\{1,2,3,4\} ?$

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Problem 39

The eccentricity of a vertex in an unrooted tree is the length of the longest simple path beginning at this vertex. A vertex is called a center if no vertex in the tree has smaller eccentricity than this vertex. In Exercises $39-41$ find every vertex that is a center in the given tree.
(GRAPH NOT COPY)

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Problem 40

The eccentricity of a vertex in an unrooted tree is the length of the longest simple path beginning at this vertex. A vertex is called a center if no vertex in the tree has smaller eccentricity than this vertex. In Exercises $39-41$ find every vertex that is a center in the given tree.
(GRAPH NOT COPY)

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Problem 41

The eccentricity of a vertex in an unrooted tree is the length of the longest simple path beginning at this vertex. A vertex is called a center if no vertex in the tree has smaller eccentricity than this vertex. In Exercises $39-41$ find every vertex that is a center in the given tree.
(GRAPH NOT COPY)

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Problem 42

Show that a center should be chosen as the root to produce a rooted tree of minimal height from an unrooted tree.

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Problem 43

Show that a tree has either one center or two centers that are adjacent.

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Problem 44

Show that every tree can be colored using two colors.

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Problem 45

The rooted Fibonacci trees $T_{n}$ are defined recursively in the following way. $T_{1}$ and $T_{2}$ are both the rooted tree consisting of a single vertex, and for $n=3,4, \ldots,$ the rooted tree $T_{n}$ is constructed from a root with $T_{n-1}$ as its left subtree and $T_{n-2}$ as its right subtree.
Draw the first seven rooted Fibonacci trees.

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Problem 46

The rooted Fibonacci trees $T_{n}$ are defined recursively in the following way. $T_{1}$ and $T_{2}$ are both the rooted tree consisting of a single vertex, and for $n=3,4, \ldots,$ the rooted tree $T_{n}$ is constructed from a root with $T_{n-1}$ as its left subtree and $T_{n-2}$ as its right subtree.
How many vertices, leaves, and internal vertices does the rooted Fibonacci tree $T_{n}$ have, where $n$ is a positive integer? What is its height?

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Problem 47

What is wrong with the following "proof" using mathematical induction of the statement that every tree with $n$ vertices has a path of length $n-1 .$ Basis step: Every tree with one vertex clearly has a path of length $0 .$ Inductive step: Assume that a tree with $n$ vertices has a path of length $n-1,$ which has $u$ as its terminal vertex. Add a vertex $v$ and the edge from $u$ to $v$ . The resulting tree has $n+1$ vertices and has a path of length $n .$ This completes the inductive step.

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Problem 48

Show that the average depth of a leaf in a binary tree with $n$ vertices is $\Omega(\log n) .$

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