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. Calculation: By carefully analyzing the structure of B k , we can infer that the number of vertices at depth J is C ( k , j ) . For example we can see that the number of vertices at various level are 1 , 4 , 6 , 4 , 1 , which are clearly C ( 4 , O ) , C ( 4 , 1 ) , C ( 4 , 2 ) , C ( 4 , 3 ) , C ( 4 , 4 ) . We can prove this by induction. Basis step: For j = O and k = O number of vertices is C ( O , O ) = 1 which is true

DISCRETE MATHEMATICS LOOSELEAF W/CONNECT

8th Edition

ISBN: 9781264309405

Author: ROSEN

Publisher: MCG

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Chapter 11, Problem 16SE

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How many vertices are there in Bk at depth j, where 0≤J≤k ? Justify your answer.

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Chapter 11, Problem 15SE

Chapter 11, Problem 17SE

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Chapter 11 Solutions

DISCRETE MATHEMATICS LOOSELEAF W/CONNECT

Ch. 11.1 - Prob. 1E Ch. 11.1 - Vhich of these graphs are trees?Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E Ch. 11.1 - Prob. 9E Ch. 11.1 - Prob. 10E

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How many vertices are there in B k at depth j , where 0 ≤ J ≤ k ? Justify your answer.

Solution Summary: The author analyzes the structure of B_k and infers that the number of vertices is C(k,j).

How many vertices are there in Bk at depth j, where 0≤J≤k ? Justify your answer.

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Chapter 11 Solutions