Binary search trees: How low can you go?

We prove that no algorithm for balanced binary search trees performing insertions and deletions in amortized time O(f(n)) can guarantee a height smaller than dlog(n + 1) + 1=f(n)e for all n. We improve the existing upper bound to dlog(n + 1) + log 2 (f(n))=f(n)e, thus almost matching our lower bound. We also improve the existing upper bound for worst case algorithms, and give a lower bound for the semi-dynamic case