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Computer science / Algorithms 1 / Chapter 1 / Problem 1.3

Algorithms | 1st Edition | ISBN: 9780073523408 | Authors: Sanjoy Dasgupta Algorithms, Christos H. Papadimitriou Algorithms, Umesh Vazirani Algorithms

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A d-ary tree is a rooted tree in which each node has at most d children. Show that any

Chapter 1, Problem 1.3

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QUESTION:

A d-ary tree is a rooted tree in which each node has at most d children. Show that any d-ary tree with n nodes must have a depth of \(\Omega(\log n / \log d)\). Can you give a precise formula for the minimum depth it could possibly have?

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QUESTION:

A d-ary tree is a rooted tree in which each node has at most d children. Show that any d-ary tree with n nodes must have a depth of \(\Omega(\log n / \log d)\). Can you give a precise formula for the minimum depth it could possibly have?

ANSWER:

Step 1 of 3

A d-ary tree with n nodes must have a depth of

A d-ary tree of height h has at most vertices. We can prove this inductively.

(a) Claim: A d-ary tree of height h has at most vertices

(b) Base Case: When the height of h of the tree is 1, the number of vertices is at most . Therefore, the statement is true when

(c) Induction hypothesis: The statement is true when the height h of the tree is

(d) Induction step: We will show that the statement is still true when . A d – ary tree of height consists of a root that has at most d children, each of which is the root of a subtree with height at most k. By the Induction hypothesis, the number of vertices in a d-ary tree of height is at most . Therefore, a tree with height has at most vertices. Hence, a tree with height K + 1 has at most vertices. We have completed the induction.

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