Question

Write an algorithm that finds the largest key in a binary search tree. Analyze your algorithm, and show the results using order notation.

Short answer

The algorithm for finding the largest key in a Binary Search Tree involves starting from the root and continuously moving to the right child until there is no right child left, which would be the largest key. The time complexity of this algorithm is O(h), where h is the height of the tree.

Step-by-step solution

Understand the Binary Search Tree (BST)

Before writing the algorithm, it's important to understand that in BST, each node has at most two children: left child and right child. Besides, each node has a key (value), and the key of the right child is always greater than its parent, while the key of the left child is always less than its parent.

Write Algorithm to Find Largest Key

In BST, the rightmost node of the tree is the node with the largest key. The algorithm will consist of moving to the right child from the root until reaching the rightmost node. The pseudocode is as follows:\n\nAlgorithm FindLargestKey(Node root)\nif root == null\n return 'Tree is empty'\nwhile root.right != null\n root = root.right\nreturn root.key\nThis algorithm starts from the root of the tree and traverses to the right child until there is no right child left.

Analyze the Algorithm

Next, perform an analysis of the algorithm to determine its time complexity. Since the algorithm has to traverse through the depth of the tree, the time complexity can be said to be linear with respect to the height of the tree. In the worst case scenario, this would correspond to a time complexity of O(h), where h is the height of the tree.