if a full binary tree has 36 internal vertices, how many leaves does it have? 1524 Searching and SortingTheorem 4.2.3: If T is a full Binary Tree with m internal vertices, then T hasт+1 leavesProof. I/ by Strong Induction on m where m E {0.. }Step 1. If m = 0, then T has a root vertex, r, but no internal vertices. Therefore,3Dr must be a leaf and must be the only vertex in T.// T has m +1 leaves.Step 2. Assume 3k EN such that if 0 <=m <= k and if T is a full Binary Treewith m internal vertices, then T has m + 1 leaves.Step 3. Suppose now that T* is a full Binary Tree with k + 1 internal vertices.// We must prove that T* has (k + 1) + 1 = k + 2 leaves.Let r* denote the root of T*; r* cannot be a leaf, so must be an internal vertex withexactly two vertices, u and v, below it in the diagram.иT2The part of the diagram from u down is full Binary Tree T1 with ml internalvertices, and the part of the diagram from v down is full Binary Tree T2 with m2internal vertices. Since every internal vertex of T is either in T1 or in T2 or r*,k+1 = m1 +m2+1where both ml and m2 are nonnegative and <= k. Since every leaf of T is either aleaf in T, or is a leaf in T2,# leaves in T = # leaves in 71 + # leaves in T2= m1 + 1+ m2 + 1= k + 2The Most Important Ideas in This Section.A Branching Diagram for an algorithm (sometimes called a “decision tree")is a tree that shows all possible sequences of operations the algorithm mightdo. Sometimes, constructing a portion of it is useful. We constructed theBranching Diagram of probes for Binary Search when n= 12 displaying allpossible sequences of probes that it might make. That diagram was a BinaryTree with 5 leaves and 7 internal vertices.(continued)

A full binary tree is a rooted tree in which each internal vertex has exactly two children. A full…

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if a full binary tree has 36 internal vertices, how many leaves does it have?

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

Problem 1PE

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