Counting Rooted Trees: The Universal Law t(n) ~ C \rho^{-n} n^{-3/2}

Abstract

Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence; for most of these a simple test can be used to quickly show that the form of the asymptotics is the same as that for the class of rooted trees: C \rho^{-n} n^{-3/2} where \rho is the radius of convergence of T.