A Short Combinatorial Proof of the Pons-Batle Identity for Counting Tree-Child Networks

Tree-child networks are a useful class of binary phylogenetic networks. The Pons--Batle identity (Pons and Batle, \textit{Scientific Reports}, 2021) states that the number $a_{n,k}$ of tree-child networks with $k$ reticulations on $n$ taxa satisfies \[ a_{n,k}=(n-k+1)a_{n,k-1} +\frac{n(2n+k-3)}{n-k}