**Problem Statement and Analysis:****Problem 1:** Binary search trees have a logarithmic search time complexity, O(lg(n)), when the tree is balanced. A balanced tree ensures no side (left or right) of the root has nodes much deeper than the other side. This concept is illustrated by a simple diagram:- The left diagram shows a "Balanced" binary tree.- The right diagram depicts an "Unbalanced" binary tree.**Objective:** Develop a formula or algorithm to measure how balanced a binary tree is.**Definitions and Formulation:**- **Height of a Node:** The height is defined as the maximum number of generations beneath a given node. Examples: - A leaf node has a height of 0. - A node with only left and/or right children and no grandchildren has a height of 1. - A node with at least one grandchild and no great-grandchildren has a height of 2. - Denote the height of a node \( n \) as \( h(n) \).- **Balance of a Node:** For a node \( n \), the balance is defined as: \[ b(n) = |h(n.left) - h(n.right)| \] Here, \( h(\text{NIL}) = -1 \) ensures the functionality when nodes do not exist.- **Balance of the Tree:** The overall balance of the tree \( T \) is given by: \[ B(T) = \max_{n \, \text{node in} \, T} b(n) \]**Algorithm Requirement:**Develop an algorithm that calculates \( B(T) \) for a binary tree \( T \). The algorithm should operate in time complexity \( O(m) \), where \( m \) is the number of nodes in the tree.

Answer: We have written the algorithm for the Balance or Unbalance Binary Search tree

blem 1: Recall that binary search trees enjoy O(ig(n)) search time complexity when the tree s fairly balanced, i.e., there are no nodes where the tree grows much deeper on the left side han the right size (and vice versa). This is well depicted by a picture: BALANCED b be 0-00 UN BALANCED n this problem, we try to come up with a formula/algorithm to measure how balanced a tree is.

blem 1: Recall that binary search trees enjoy O(ig(n)) search time complexity when the tree s fairly balanced, i.e., there are no nodes where the tree grows much deeper on the left side han the right size (and vice versa). This is well depicted by a picture: BALANCED b be 0-00 UN BALANCED n this problem, we try to come up with a formula/algorithm to measure how balanced a tree 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

See similar textbooks

Similar questions

Artificial Inteligence
Question # 01: Consider the searching problem: Input: A sequence of n numbers A = (a1, a2, a3, an ) a value V. Output: An index i such that v = A[i] or the special value NIL if v does not appear in A. Write pseudocode for linear search, which scans through the sequence, looking for v. Using a loop invariant, prove that your algorithm is correct. Make sure that your loop invariant fulfills the three necessary properties.
BACKTRACKING
Using an in-order traversal it is quite easy to output all the data in a binary search tree in order. Design an algorithm which when given a binary search tree and two numbers x and y outputs all the data items z in order with the property that x
9T(플) + 0(n) 1 T(n) = 3 Find the upper bound of this recurrence equation using (i) the master method (ii) the recursion tree method
Kruskal's Algorithm: Kruskal's algorithm is a minimum-spanning-tree algorithm which finds an edge of the least possible weight that connects any two trees in the forest. This algorithm is a Greedy Algorithm. The steps to find a MST using this algorithm are as follows: Sort all the edges in non-decreasing order of their weight. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If cycle is not formed, include this edge. Else, discard it. Repeat step2 until there are (V-1) edges in the spanning tree. Example with 10 vertices and 20 weighted edges:
b. Given a BST whose root node is represented by the node root. Write a function findroots(Node * root) to traverse the BST and print the number of total parent (atleast have a child node) nodes of the BST. N.B: Print order doesn't matter Fig 1: An example Binary Search Tree. 5 For example: In this figure, there are total 4 parents
3) Use recursion Tree to determine a good asymptotic upper bound on the recurrence T(n) = 4T (" + 2) + n Use the substitution method to verify your answer.
If you decide to print the content of binary search tree in acceding order, you have to choose the: a. RLV traversal b. in-order traversal c. pre-order traversal d. post-order traversal