On the Asymptotic Growth of the Number of Tree-Child Networks

TL;DR

This paper precisely characterizes the asymptotic growth of the number of tree-child networks with n leaves, refining previous estimates by identifying the main growth term and its constant factors.

Contribution

It provides a complete asymptotic formula for counting tree-child networks, using a bijective approach and advanced asymptotic methods to improve prior results.

Findings

Number of tree-child networks grows like \[ \Theta\left(n^{-2/3}e^{a_1(3n)^{1/3}}\left(\frac{12}{e^2}\right)^{n}n^{2n}\right)\]

Identifies the constant factors and the dominant exponential growth term.

Uses a bijective mapping to translate the problem into a word-based combinatorial problem.

Abstract

In a recent paper, McDiarmid, Semple, and Welsh (2015) showed that the number of tree-child networks with $n$ leaves has the factor $n^{2n}$ in its main asymptotic growth term. In this paper, we improve this by completely identifying the main asymptotic growth term up to a constant. More precisely, we show that the number of tree-child networks with $n$ leaves grows like \[ \Theta\left(n^{-2/3}e^{a_1(3n)^{1/3}}\left(\frac{12}{e^2}\right)^{n}n^{2n}\right), \] where $a_1=-2.338107410\cdots$ is the largest root of the Airy function of first kind. For the proof, we bijectively map the underlying graph-theoretical problem onto a problem on words. For the latter, we can find a recurrence to which a recent powerful asymptotic method of Elvey Price, Fang, and Wallner (2019) can be applied.