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if a binary tree has height 5, what is the maximum number of leaves it can have?

if a binary tree has height 5, what is the maximum number of leaves it can have?

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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Based on Theorem 4.4.1 P.163, if a binary tree has height 5, what is the maximum number of leaves it can have?

4.4 Binary Trees with (at Least) n! Leaves 163 4.4 Binary Trees with (at Least) n! Leaves A Binary Tree is a tree T with a certain distinguished vertex, r, called the root of T and placed at the top of the diagram, and where each vertex, v, of T has at most two vertices directly below it in the diagram. A leaf is a vertex with no vertices below it; the other (non-leaf) vertices are known as internal vertices. For any vertex, v, there is a unique path from it back to the root r. // up the diagram and down the tree The length of that path is the number of edges traversed and is called the height of vertex v and denoted by h(v). The height of the tree T equals the height of the highest vertex in T. // so h(r) = 0 %3D // the highest leaf Theorem 4.4.1: A Binary Tree with height h has at most 2" leaves. Proof. I/ by Strong Induction on h where h e {0.. } Step 1. If T has height h = 0, there can be no vertex in J other than the root, r. Then, r must be a leaf and so T has exactly 1 = 2" leaves. Step 2. Assume 3q >= 0 such that for any k where 0 <= k <= q, any Binary Tree with height k has at most 2* leaves. Step 3. Suppose T* is some particular Binary Tree with height q + 1 rooted at vertex r. // We must prove that T* has at most 2ª+1 leaves.
Transcribed Image Text:4.4 Binary Trees with (at Least) n! Leaves 163 4.4 Binary Trees with (at Least) n! Leaves A Binary Tree is a tree T with a certain distinguished vertex, r, called the root of T and placed at the top of the diagram, and where each vertex, v, of T has at most two vertices directly below it in the diagram. A leaf is a vertex with no vertices below it; the other (non-leaf) vertices are known as internal vertices. For any vertex, v, there is a unique path from it back to the root r. // up the diagram and down the tree The length of that path is the number of edges traversed and is called the height of vertex v and denoted by h(v). The height of the tree T equals the height of the highest vertex in T. // so h(r) = 0 %3D // the highest leaf Theorem 4.4.1: A Binary Tree with height h has at most 2" leaves. Proof. I/ by Strong Induction on h where h e {0.. } Step 1. If T has height h = 0, there can be no vertex in J other than the root, r. Then, r must be a leaf and so T has exactly 1 = 2" leaves. Step 2. Assume 3q >= 0 such that for any k where 0 <= k <= q, any Binary Tree with height k has at most 2* leaves. Step 3. Suppose T* is some particular Binary Tree with height q + 1 rooted at vertex r. // We must prove that T* has at most 2ª+1 leaves.
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