The minimum number of elements in a heap of height h is

The minimum number of elements in a heap of height h is 2^h.

Here's the step by step explanation:

1. A heap is a complete binary tree. In a complete binary tree, all levels are completely filled except possibly for the last level which is filled from left to right.

2. The height of a heap is the number of edges on the longest path from the root node to a leaf node.

3. In a heap of height h, the minimum number of elements is achieved when there is only one node at the height h (i.e., the last level of the tree).

4. For a heap of height h, there are 2^0 nodes at height 0 (the root), 2^1 nodes at height 1, 2^2 nodes at height 2, and so on, up to 2^(h-1) nodes at height h-1.

5. At height h, there is at least one node.

6. Therefore, the minimum number of nodes in a heap of height h is the sum of the number of nodes at each height, which is 2^0 + 2^1 + 2^2 + ... + 2^(h-1) + 1.

7. This sum is a geometric series with first term 1, common ratio 2, and h terms. The sum of a geometric series is given by the formula a * (r^n - 1) / (r - 1), where a is the first term, r is the common ratio, and n is the number of terms.

8. Substituting the values a = 1, r = 2, and n = h into the formula, we get 1 * (2^h - 1) / (2 - 1) = 2^h - 1.

9. But we have to add the one node at height h, so the minimum number of nodes in a heap of height h is 2^h - 1 + 1 = 2^h.