Binary search trees: How low can you go?

Abstract

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 ⌈log(n + 1) + 1/f(n)⌉ for all n. We improve the existing upper bound to ⌈log(n + 1) + log2 (f(n))/f(n)⌉, 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.