Question 1

How many nonisomorphic trees are there with four vertices? Draw them.

Accepted Answer

There are two nonisomorphic unrooted trees with four vertices, as shown.

Question 2

(a) Suppose that a full 3-ary tree has 100 internal vertices. How many leaves does it have? (b) Suppose that a full 4-ary tree has 100 leaves. How many internal vertices does it have?

Accepted Answer

(a) A full 3-ary tree with 100 internal vertices has l = (3 − 1) · 100 + 1 = 201 leaves. (b) A full 4-ary tree with 100 leaves has i = (100 − 1)/(4 − 1) = 33 internal vertices.

Question 3

Perform a preorder, inorder, and postorder traversal of the rooted tree below.

Accepted Answer

The preorder traversal is a, b, e, f, j, k, c, d, g, h, i; the inorder traversal is $e , b , j , f , k , a , c , g , d , h , i$ and the postorder traversal is e, j, k, f, b, c, g, h, i, d, a.

Question 4

Construct an expression tree for $( 3 + x ) - 5 \cdot y$ and write this expression is preﬁx form and postﬁx form.

Accepted Answer

The following tree represents the expres

Question 5

Use backtracking to ﬁnd a sum of integers in the set {18, 19, 23, 25, 31} that equals 44.

Accepted Answer

We build the followi

Question 6

Use a depth-ﬁrst search to ﬁnd a spanning tree of the following graph. Start at the vertex a, and use alphabetical order.

Accepted Answer

The edges produced b

Question 7

Is the code $\mathrm { A } : 11 , \mathrm {~B} : 10 , \mathrm { C } : 0$a prefix code?

Accepted Answer

This is a preﬁx code since the code for

Question 8

A tree has 99 edges. How many vertices does it have?

Accepted Answer

If a tree has e edge

Question 9

Is the code A: 111, B: 101, C: 011, D: 010, E: 10, F: 1101 a preﬁx code?

Accepted Answer

This is not a preﬁx

Question 10

Form a binary search tree from the words of the sentence This test is not so diﬃcult, using alphabetical order, inserting words in the order they appear in the sentence.

Accepted Answer

The follow

Question 11

Which of the following graphs are trees ? Explain your answers.
a)
 
b)
 
C)

Accepted Answer

The graph in part (a) is not connected,

Question 12

Find a minimum spanning tree in the following weighted graph using Prim's algorithm.

Accepted Answer

Prim's algorithm adds the edges {d, e} o

Question 13

Use Prim's algorithm to ﬁnd a minimum spanning tree for the following weighted graph. Use alphabetical

Accepted Answer

Prim's algorithm adds the edge

How many nonisomorphic trees are there with four vertices? Draw them.

(a) Suppose that a full 3-ary tree has 100 internal vertices. How many leaves does it have? (b) Suppose that a full 4-ary tree has 100 leaves. How many internal vertices does it have?

Perform a preorder, inorder, and postorder traversal of the rooted tree below.

Construct an expression tree for $( 3 + x ) - 5 \cdot y$ and write this expression is preﬁx form and postﬁx form.

Use backtracking to ﬁnd a sum of integers in the set {18, 19, 23, 25, 31} that equals 44.

Use a depth-ﬁrst search to ﬁnd a spanning tree of the following graph. Start at the vertex a, and use alphabetical order.

Is the code $\mathrm { A } : 11 , \mathrm {~B} : 10 , \mathrm { C } : 0$ a prefix code?

A tree has 99 edges. How many vertices does it have?

Is the code A: 111, B: 101, C: 011, D: 010, E: 10, F: 1101 a preﬁx code?

Form a binary search tree from the words of the sentence This test is not so diﬃcult, using alphabetical order, inserting words in the order they appear in the sentence.

Which of the following graphs are trees ? Explain your answers. a) b) C)

Find a minimum spanning tree in the following weighted graph using Prim's algorithm.

Use Prim's algorithm to ﬁnd a minimum spanning tree for the following weighted graph. Use alphabetical

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

Exam 11: Trees

How many nonisomorphic trees are there with four vertices? Draw them.

(a) Suppose that a full 3-ary tree has 100 internal vertices. How many leaves does it have? (b) Suppose that a full 4-ary tree has 100 leaves. How many internal vertices does it have?

Perform a preorder, inorder, and postorder traversal of the rooted tree below.

Construct an expression tree for (3+x)−5⋅y( 3 + x ) - 5 \cdot y(3+x)−5⋅y and write this expression is preﬁx form and postﬁx form.

Use backtracking to ﬁnd a sum of integers in the set {18, 19, 23, 25, 31} that equals 44.

Use a depth-ﬁrst search to ﬁnd a spanning tree of the following graph. Start at the vertex a, and use alphabetical order.

Is the code A:11, B:10,C:0\mathrm { A } : 11 , \mathrm {~B} : 10 , \mathrm { C } : 0A:11, B:10,C:0 a prefix code?

A tree has 99 edges. How many vertices does it have?

Is the code A: 111, B: 101, C: 011, D: 010, E: 10, F: 1101 a preﬁx code?

Form a binary search tree from the words of the sentence This test is not so diﬃcult, using alphabetical order, inserting words in the order they appear in the sentence.

Which of the following graphs are trees ? Explain your answers. a) b) C)

Find a minimum spanning tree in the following weighted graph using Prim's algorithm.

Use Prim's algorithm to ﬁnd a minimum spanning tree for the following weighted graph. Use alphabetical

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

Construct an expression tree for $( 3 + x ) - 5 \cdot y$ and write this expression is preﬁx form and postﬁx form.

Is the code $\mathrm { A } : 11 , \mathrm {~B} : 10 , \mathrm { C } : 0$ a prefix code?