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

Abstract

Combinatorial classes ${\cal T}$ that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series ${\bf T}(z)$ 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 ${\bf T}$.