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

TL;DR

This paper studies the enumeration of leaf-labelled increasing trees with arbitrary node degrees, validating asymptotic formulas and comparing their growth to other tree types, with implications for population genetics models.

Contribution

It provides a rigorous validation of asymptotic formulas for counting such trees and compares their growth to binary, ternary, and quaternary trees.

Findings

Validated asymptotic enumeration formulas for arbitrary-degree leaf-labelled increasing trees.

Compared growth rates of these trees to binary, ternary, and quaternary trees.

Connected combinatorial results to population-genetics models.

Abstract

We consider the counting problem of the number of \textit{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-labelled increasing trees to that of binary, ternary and quaternary trees.