Here is a sketch of the proof that an AVL tree with N keys has height h = O(log(N)).

Let f(h) be the minimum number of keys in an AVL tree of height h.

(For example, try to draw AVL trees of heights 1, 2, and 3 with the smallest number of keys.)

      f(-1) = 0
      f(0) = 1
      f(1) = 2
   

For h >= 1,