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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5. Suppose we have a binary tree of height h. Prove that the number of leaves of this binary tree is at most 2" (that is 2 to the power of h).
Topic: Binary Tree Graph
Given a binary tree, find if it is height balanced or not. A tree is height balanced if difference between heights of left and right subtrees is not more than one for all nodes of tree. A height balanced tree 1 I 10 39 . 5 An unbalanced tree 1 I 10 I 5 Example 1: Input: 1 2 1 Output: 0
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 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 recur- sively.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 binary tree is a rooted tree with no more than two offspring per node. Show that in a binary tree, the number of nodes that produce two offspring is precisely one fewer than the number of leaves.
Construct a binary tree from the given preorder and inorder traversals. Preorder: GBQACK EPDERH Inorder QBKCFAGPEDHR
Please a personal solution, not from the internet, correct and clearAlgorithms
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.}
[Note: Assume height of leaf node = 0 and height of root node = h. For (a) and (b), you need to explain the number of nodes at each level of the binary tree. For answering part (c), use the results in (a) and (b)]
If T is a binary tree with 100 vertices, what is its minimum height?
Prove that any binary tree of height h (where the empty tree is height 0, and a tree witha single node is height 1) has between h and 2h − 1 nodes, inclusive. A binary tree is onein which every node has at most three edges (at most one to the ’parent’ and two to the’children.’)
if T is a complete 4-ary tree (rooted tree where every internal nonleaf vertex has 4 children. if x is the number of leaves of T a.how many internal nodes does it have? b.how many total vertices? c. minimum height possible? d. maximum height possible? e. if Y is the height of T then what are the minimum and maximum number of vertices?

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