Rank of root in AVL tree
First of all, it's not $2^{h-2}$, it's $2^h$.

The number of nodes $n$ in a full binary tree, is at most $n=2^{h+1}-1$, where $h$ is the height of the tree. A tree consisting of only a root node has a height of 0.

Straight outta wiki https://en.wikipedia.org/wiki/AVL_tree