Exploring binary trees and other simple trees

The average height of a binary tree With n internal nodes is shown to be asymptotic to 2√πn. More generally, the average height of a tree in a simple family S with n nodes is asymptotic to c(S) √πn where c(S) is a number (usually algebraic) which can be explicitly determined from S. These results are achieved by means of a detailed singularity analysis of corresponding generating functions.