Explanation Given: B k Concept used: The binomial trees B k , 1 = 0 , 1 , 2 , ... , are ordered rooted trees defined recursively: Basis step: The binomial tree B 0 is the tree with a single vertex. Recursive step: Let k be a nonnegative integer. To construct the binomial tree B k + 1 , add a copy of B k to a second copy of B k by adding an edge that makes the root of the first copy of B k the leftmost child of the root of the second copy of B k Principle of mathematical induction is used to verify mathematical and logical expression. Proof: We shall prove using mathematical induction that root and the left most child of the root have the highest degree in B k and degree will be k and every other vertex in B k has degree less than k . Basis step: - This is true for B 0 and B 1 . Inductive hypothesis: - Assume that root and the left most child of the root have the highest Degree in B k and degree will be k . By definition B k + 1 is formed by adding an edge to the root of one B k and makes another one B k as its left Child

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

8th Edition

ISBN: 9781259676512

Author: Kenneth H Rosen

Publisher: McGraw-Hill Education

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Question

Chapter 11, Problem 18SE

To determine

To show:

The vertex of largest degree in Bk is the root.

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Chapter 11, Problem 17SE

Chapter 11, Problem 19SE

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