Explanation Given: B-tree of degree k . Concept used: A B − Tree of degree k is a rooted tree such that all its leaves are at the same level, its root has at least two and at most k children unless it is a leaf, and every internal vertex other than the root has at least ⌈ k 2 ⌉ , but no more than k , children. Computer files are often accessed expeditiously once B − trees are accustomed to represent them. Calculation: Suppose the height of the tree is h . Maximum leaves occur at level h when all the internal nodes Have k children at level 1 k nodes and each of the nodes having k children like that the structure Repeats. So at level h there will be k h leaves. That is, we can have at most k h leaves which is given as n . That is, n ≤ k h So, h ≥ log k n This forms the lower bound to the height of the tree. Now, again the tree will be having minimum number of leaves if all the internal nodes are having ⌈ k 2 ⌉ Nodes and root is having 2 nodes

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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Chapter 11, Problem 12SE

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An upper bound and a lower bound for the height of a B-tree of degree k with n leaves.

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Chapter 11, Problem 11SE

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Chapter 11 Solutions

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

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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An upper bound and a lower bound for the height of a B-tree of degree k with n leaves.

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An upper bound and a lower bound for the height of a B-tree of degree k with n leaves.

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Chapter 11 Solutions