(a) Explanation Description: Given the length from root to the terminal vertex is h . Thus, there are h levels in the tree. Note that there is only one vertex at the level 0. Then there are at most K 1 = K vertices at the level 1. Also, there are at most K 2 vertices at the second level. In a similar fashion it can be said that there are at most K n vertices at the level n . Suppose that there are n vertices in the tree T . Thus, the below inequality can be derived (b) To determine To prove: That if a vertex has k children, then T has at least h + k vertices.

5th Edition

ISBN: 9780134689562

Author: Dossey, John A.

Publisher: Pearson,

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Question

Chapter 5, Problem 30SE

(a)

To determine

To prove: That the rooted tree T has at most kh+1−1k−1 vertices if each vertex of T has k≥2 children and the length of the longest path from the root to the terminal vertex is h.

(b)

To determine

To prove: That if a vertex has k children, then T has at least h+k vertices.

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Chapter 5, Problem 29SE

Chapter 5, Problem 31SE

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Chapter 5 Solutions

Discrete Mathematics

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Ch. 5.1 - Seven farming communities in Iowa want to develop...Ch. 5.1 - As few trails as possible are to be built between...Ch. 5.1 - A farmer needs to irrigate the fields in which his...Ch. 5.1 - Draw a graph that is not a tree for which the...Ch. 5.1 - Prob. 15E Ch. 5.1 - What is the smallest number of edges in a...Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - There are two saturated hydrocarbons of the type...Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - In order to count the number of distinct trees...Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - In order to count the number of distinct trees...Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - 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Knowledge Booster

To prove: That the rooted tree T has at most k h + 1 − 1 k − 1 vertices if each vertex of T has k ≥ 2 children and the length of the longest path from the root to the terminal vertex is h .

To prove: That the rooted tree T has at most kh+1−1k−1 vertices if each vertex of T has k≥2 children and the length of the longest path from the root to the terminal vertex is h.

To prove: That if a vertex has k children, then T has at least h+k vertices.

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Chapter 5 Solutions