1. A complete binary tree of depth d is a tree where every node has two children, except for the nodes at depth d, whichhave no children. Such a tree has 2' nodes at depth j, where j = 0, 1,d.What is the minimal size of a vertex cover of a complete binary tree of depth 2m + 1, where m is a nonnegative integer?

Answered: 1. A complete binary tree of depth d is…

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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A binary tree is \emph{full} if every non-leaf node has exactly two children. For context, recall that we saw in lecture that a binary tree of height $h$ can have at most $2^{h+1}-1$ nodes and at most $2^h$ leaves, and that it achieves these maxima when it is \emph{complete}, meaning that it is full and all leaves are at the same distance from the root. Find $\nu(h)$, the \emph{minimum} number of leaves that a full binary tree of height $h$ can have, and prove your answer using ordinary induction on $h$. Note that tree of height of 0 is a single (leaf) node. \textit{Hint 1: try a few simple cases ($h = 0, 1, 2, 3, \dots$) and see if you can guess what $\nu(h)$ is.}
Assuming that a tree with a single node has a height of 1, what is the maximum number of nodes that may exist in a binary tree with a height of 5?
prove the following a.In a tree all nodes, except the leaves will have betweenness centrality greater than zero b.When eliminating the nodes of a tree to form the Prufer code, the vertex with the highest label will not be deleted
A complete binary tree is a graph defined through the following recur- sive definition. Basis step: A single vertex is a complete binary tree.Inductive step: If T1 and T2 are disjoint complete binary trees with roots r1, r2, respectively, the graph formed by starting with a root r, and adding an edge from r to each of the vertices r1,r2 is also a complete binary tree.The set of leaves of a complete binary tree can also be defined recursively.Basis step: The root r is a leaf of the complete binary tree with exactly one vertex r.Inductive step: The set of leaves of the tree T built from trees T1, T2 is the union of the sets of leaves of T1 and the set of leaves of T2. The height h(T ) of a binary tree is defined in the class.Use structural induction to show that L(T), the number of leaves of a complete binary tree T , satisfies the following inequality L(T) ≤ 2^h(T).
A tree that is rooted and has a maximum of two children per node is commonly known as a binary tree. The purpose of this demonstration is to establish that the quantity of nodes present in a binary tree that are accountable for generating two children is precisely one unit less than the quantity of leaves in the same tree.
A binary tree is a rooted tree that has no more than two children per node. Show that the number of nodes in a binary tree that produce two offspring is precisely one fewer than the number of leaves in that tree.
Given that a tree with a single node has a height of one, what is the maximum number of nodes that may be included in a balanced binary tree with a height of five?
A binary tree is a rooted tree in which each node may produce no more than two children. Prove that in any binary tree, the number of nodes that produce two offspring is exactly one less than the number of leaves.
Given that a tree with a single node has a height of one, what is the maximum number of nodes that may be included in a balanced binary tree that has a height of 5?
What is the maximum number of nodes in a balanced binary tree of height 5, assuming that a tree with only a single node has height 1?

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