# What are the minimum and maximum numbers of elements in a heap of height h?

I came across the question:

What are the minimum and maximum numbers of elements in a heap of height $$h$$?

To which I came up with this theory: $$\sum_{i=0}^{h-1} 2^i = 2^h-1$$

$$2^h-1$$ is the internal nodes and that is understood according to the understood fact. But because nowhere except CLRS mentions heaps to be a Nearly Complete Binary Tree everywhere it is mentioned as a Complete Binary Tree.

The maximum number of elements can be easily computed: $$\sum_{i=0}^{h} 2^i = 2^{h+1}-1$$

But I cannot get the point of computing the minimum number of elements: Should it be: $$2^{h}+1$$ for $$0$$ or $$2$$ children property

Or should it be: $$2^{h}$$ for $$0$$ or $$1$$ children property

Reference2: HeapSort

Thank you.

A heap of height $$h$$ is complete up to the level at depth $$h-1$$ and needs to have at least one node on level $$h$$.
Therefore the minimum total number of nodes must be at least $$\sum_{i=0}^{h-1} 2^i + 1 = 2^{h}-1 + 1= 2^h.$$, and this tight since an heap with $$2^h$$ nodes has height $$h$$.