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

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 $$\mathrm{\Theta }\left(n^{-2\mathrm{/}3}{e}^{a_1(3n{)}^{1\mathrm{/}3}}{\left(\frac{12}{e^2}\right)}^n{n}^{2n}\right),$$ 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.