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math.CO2026
A Short Combinatorial Proof of the Pons-Batle Identity for Counting Tree-Child Networks
Hao Yu, Louxin Zhang
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}…
math.CO2026
A cube-root phase transition in tree-child networks and the enumeration threshold for galled networks
Michael Fuchs, Hao Yu
We prove two surprising results about phylogenetic networks. First, we show that the structure of tree-child networks with leaves and reticulation nodes undergoes a sharp p…