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