On the enumeration of leaf-labeled increasing trees with arbitrary node-degree

Abstract

We consider the counting problem of the number of leaf-labeled increasing trees, where internal nodes may have an arbitrary number of descendants. The set of all such trees is a discrete representation of the genealogies obtained under certain population-genetical models such as multiple-merger coalescents. While the combinatorics of the binary trees among those are well understood, for the number of all trees only an approximate asymptotic formula is known. In this work, we validate this formula up to constant terms and compare the asymptotic behavior of the number of all leaf-labeled increasing trees to that of binary, ternary and quaternary trees.