1. (50 points) Show that the tree height of a height-balanced binary search tree with n nodes is 0(log n). (Hint: Let T(h) denote the fewest number of nodes that a height-balanced binary search tree of height h can have. Express T(h) in terms of T (h-1) and T(h-2). Then, find a lower bound of T(h) in terms of T (h-2). Finally, express the lower bound of T (h) in terms of h.)