The height of the latest common ancestor of two randomly chosen leaves from a (sub-)critical Galton–Watson tree

Abstract

Take a complete (sub-)critical Galton–Watson branching tree with finitely many leaves almost surely. Picking two distinct leaves at random, we ask for the height (as measured from the root) of their latest common ancestor. Upon conditioning on the first branching event we compute the distribution of this height, with calculations explicit in the case of a binary tree.