Suppose that the splits at every level of quicksort are in the proportion of 1 – a and a, where 0 < a <= ½ is a constant. Show that the minimum depth of a leaf node in the recursion tree is approximately (–lg n) / (lg a) and the maximum depth is approximately (– lg n) / lg(1– a).

Let T be a binary search tree with n nodes, in which n > 0. When T is linear, the search algorithm makes key comparisons, in the unsuccassful case.

Solve using the recursion tree T(n) = 4T(n-1) + c, where c is a positive constant, and T(0)=0.

A Binary Search Tree (BST) can be used to efficiently implement a sorted set. It stores uniquevalues and offers an efficient membership method (contains).A binary search tree (BST) is a binary tree with the additional binary search tree property.The binary search tree property states that for every node n in the tree1. n.data is greater than the data value of all nodes in the sub-tree rooted with n.left.2. n.data is less than the data value of all nodes in the sub-tree rooted with n.right.As we are implementing a set, all data (strings) in the tree will be unique. Every node in the treewill have its own distinct string. Strings will be compared using the canonical way (as defined bythe compareTo() method in the String class).For example, the binary tree on the left (below) IS a binary search tree and the one on the rightis NOT.You will implement several methods in the BinaryTree and BST classes. The BST class mustextend the BinaryTree class. Methods that you must complete (implement or…

Given an m-by-n chocolate bar, you need to break it into mn 1-by-1 pieces. You can break a bar only in a straight line, and only one bar can be broken at a time. Design an algorithm that solves the problem with the minimum number of bar breaks. What is this minimum number? Breaking the chocolate bar can be represented by a binary tree.