{
  "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 $a_{n,k}$ 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}a_{n-1,k}. \\] In this paper, we present a short combinatorial proof of this identity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04979",
  "author_total": 2,
  "authors": [
    {
      "name": "Hao Yu"
    },
    {
      "name": "Louxin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jmhn0p7exd2q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04979",
  "fulltext_url": null,
  "id": "8jmhn0p7exd2q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04979",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "q-bio.PE"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7211879296",
  "pdf_url": "https://arxiv.org/pdf/2609.04979",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jmhn0p7exd2q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04979",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04979"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.PE",
      "subfield": "evolutionary-biology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Short Combinatorial Proof of the Pons-Batle Identity for Counting Tree-Child Networks",
  "title_text": "A Short Combinatorial Proof of the Pons-Batle Identity for Counting Tree-Child Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:19Z",
  "url": "https://research.draconitic.com/paper/8jmhn0p7exd2q",
  "venue": "arXiv",
  "versions": []
}