# Rank of root in AVL tree

Find a function that bounds from above and below (asymptotic) the rank of the root r in an AVL tree,
i.e. find a function $$f(n)$$ so there exists a constant $$c>0$$ that for every AVL tree with $$n$$ nodes so $$c * f(n)\le rank(r)\le n-c*f(n)$$.

$$rank(r)$$ gets his maximum value when his sub-left tree contains maximal number of nodes and because it's an AVL tree, for tree with height $$h$$ the sub left tree will be with height of $$h-1$$ and the number of nodes will be maximal when the sub tree is a full tree with $$2^{h-2}-1$$ nodes hence we get:
$$rank(r)\leq 2^{h-2}-1+1=2^{h-2}$$

On the other hand, $$rank(r)$$ gets his minimal value when his sub left tree contains a minimal number of nodes and it happens when the sub left tree is a fibonacci tree with height of $$h-2$$, therefore
$$rank(r) \geq \phi ^{h-2}+1$$.

Now I need to find f as a function of the number of nodes $$n$$ and not the height $$h$$. Am I in the right direction?
Would apricciate any help how to continue, Thanks.

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.