I studying the related question.
but it's not so general.
In-fact, We want to know with N keys, how many different AVL Tree, we can make?
we know with N=1 key is 1 AVL Tree. with N=2 key we have 2 different AVL Tree, but in general we can make any recurrence formula? for example for N=4, N=5 and so on.
Let $a_{n,h}$ denote the number of AVL trees with $n$ nodes and height $h$. It is straightforward to get a recurrence for $a_{n,h}$:
$$a_{n,h} = \sum_{k=1}^n \bigl(a_{k-1,h-1}a_{n-k,h-1} + a_{k-1,h-1}a_{n-k,h-2} + a_{k-1,h-2}a_{n-k,h-1}\bigr),~ n\geq h > 1,$$
with the initial conditions $a_{n,h} = 0$, if $h>n$ or $h\in\{0,1\}, n\neq h$, and $a_{0,0} = a_{1,1} = 1$.
The numbers you are looking for then are $a_n = \sum_h a_{n,h}$.
Unfortunately there apparently is no known closed form for this sequence. The OEIS has a list of the first 1000 terms and maple + mathematica code to compute further ones (via the recursion).
