(AVL Trees) What is the maximum possible difference between the number of nodes in the root node's subtrees?
I assume that we start labeling heights from $0$.

Actually your approach is right. You must consider the left-side (or right-side) sub-tree with the least possible amount of nodes and the right-side (or left-side) sub-tree with the most possible amount of nodes, and both sub-trees must be AVL trees too. For this purpose, we fix the height of left-side sub-tree to $h-2$ (doing so, the bottommost node of this sub-tree would be at height $h-1$ of the main tree) and we fix the height of right-side sub-tree to $h-1$. Now, we should answer following questions:


What is the minimum number of nodes in an AVL tree of height $h-2$?
What is the maximum number of nodes in an AVL tree of height $h-1$?


Answer to question 1: By recursions and assuming that $n(h)$ is the minimum number of nodes of an AVL tree of height $h$ we can easily get:
$$
\begin{array}{rl}
n(h) &amp;=&amp; n(h-1)+n(h-2)+1\\
n(1)&amp;=&amp;2\\
n(0)&amp;=&amp;1
\end{array}
$$

Answer to question 2: In this case, we're dealing with a perfect tree (which is an AVL tree too) and we must find the number of nodes in a perfect tree of height $h-1$.

So, the answer to your question would be:
$$
2^h - 1 - n(h-2)
$$

As an example, if $h=2$, this formula returns $2$, which is not the value you have got, and I think that's because you start labeling heights from $1$.