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PreprintCC BY 4.0arXiv

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

  • Hao Yu
  • Louxin Zhang
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Abstract

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 an,k of tree-child networks with k reticulations on n taxa satisfies an,k=(n−k+1)an,k−1 +(n(2n+k−3))/(n−k)an−1,k. In this paper, we present a short combinatorial proof of this identity.

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Cite

BibTeXRIS
BibTeX
@misc{yu2026short,
  title = {{A Short Combinatorial Proof of the Pons-Batle Identity for Counting Tree-Child Networks}},
  author = {Hao Yu and Louxin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04979},
  eprint = {2609.04979},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04979},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Short Combinatorial Proof of the Pons-Batle Identity for Counting Tree-Child Networks
AU  - Hao Yu
AU  - Louxin Zhang
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04979
UR  - https://arxiv.org/abs/2609.04979
AB  - 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 +(n(2n+k−3))/(n−k)a_n−1,k. In this paper, we present a short combinatorial proof of this identity.
ER  -