Binary Search Tree (BST) Key Storage Capacity

How many keys can be stored in a Binary Search Tree (BST) with k levels?

a) k

b) 2ᵏ

c) 2ᵏ⁺¹ - 1

d) k²

Answer:

The maximum number of keys that can be stored in a Binary Search Tree (BST) with k levels is represented by the formula 2^k - 1, which corresponds to option c) 2ⁿ¹ - 1.

Explanation:

The question asks about the maximum number of keys that can be stored in a Binary Search Tree (BST) with k levels. The answer to this question is c) 2ⁿ¹ - 1, because in a BST with k levels, the number of keys is based on the concept that each level can potentially store twice the number of keys as the preceding level (except the first level which starts with a single key).

At the first level (level 0), there is 1 key. At the second level, there are 2 keys. This pattern continues such that at level k, the number of keys is 2ⁿ (if every possible node is filled). To find the total number of keys stored in k levels, you would sum up the number of keys at each level:

1 + 2 + 4 + 8 + ... + 2ⁿ¹ = 2ⁿ - 1

This is a geometric series with the sum given by the formula for the sum of the first n terms of a geometric progression where the first term is 1 and the common ratio is 2.