How to prove that in an AVL tree with height h, the depth of every leaf node is at least $\lceil h/2 \rceil$
Denote by $d(h)$ the minimum height of a leaf in an AVL tree of height $h$. One subtree of the root necessarily has height $h-1$, and the other one has height either $h-2$ or $h-1$ by the defining property of AVL trees. Therefore
$$
d(h) = \min(d(h-1),d(h-2)) + 1.
$$
Also, one checks that $d(0) = d(1) = 0$ (if one measures height as the maximum number of edges in a root-to-leaf path). The recurrence makes it clear that $d$ is monotone, and so $d(h) = d(h-2) + 1$. From here it's easy to calculate $d(h) = \lfloor h/2 \rfloor$.