{
  "abstract": "We develop a unified framework for the exact enumeration and asymptotic analysis of the three most studied classes of phylogenetic networks: tree-child (TC), reticulation-visible (RV) and orchard networks, whose cardinalities satisfy the strict ordering $|\\mathrm{TC}_{\\ell,k}|<|\\mathrm{RV}_{\\ell,k}|<|\\mathrm{Orch}_{\\ell,k}|$ for reticulation number $k\\geq2$ (with $\\mathrm{TC}\\subsetneq\\mathrm{RV}$ and $\\mathrm{TC}\\subsetneq\\mathrm{Orch}$, while $\\mathrm{RV}$ and $\\mathrm{Orch}$ are incomparable as sets). Using the Chang--Fuchs structural theorem, we derive a two-level master functional equation for the RV bivariate generating function and obtain exact closed-form identities for the differences $Δ_k(\\ell):=|RV_{\\ell,k}|-|TC_{\\ell,k}|$ for $k=2,3$, with the asymptotic universality $Δ_k(\\ell)/|TC_{\\ell,k}|\\sim k!/\\ell$. For orchard networks, we prove a \\emph{universal hypergeometric law} that resolves the exact enumeration problem for all $\\ell$: the column generating function $F_\\ell(v)$ is rational with denominator $D_\\ell(v)=\\prod_{j=2}^\\ell X_j(v)$, where \\[ X_\\ell(v) = \\sum_{k=0}^{\\lfloor\\ell/2\\rfloor}(-1)^k\\, \\frac{\\ell!}{(\\ell-2k)!\\,k!}\\,v^k \\] is the matching polynomial of the complete graph $K_\\ell$ and a rescaled Jacobi polynomial. This immediately resolves the intractable $\\ell=9$ case: $D_9$ has degree 20, dominant growth rate $\\approx40.73$, and all spectral roots are positive real. A complete enumeration table is provided extending the published data of Cardona, Ribas and Pons.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.24325",
  "author_total": 1,
  "authors": [
    {
      "name": "Josep Batle"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/af0qj3z860hgc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.24325",
  "fulltext_url": null,
  "id": "af0qj3z860hgc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.24325",
  "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": "W7165788579",
  "pdf_url": "https://arxiv.org/pdf/2606.24325",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/af0qj3z860hgc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2606.24325",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.24325"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.PE",
      "subfield": "evolutionary-biology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact Enumeration of Phylogenetic Networks: The Tree-Child, Reticulation-Visible and Orchard Hierarchy",
  "title_text": "Exact Enumeration of Phylogenetic Networks: The Tree-Child, Reticulation-Visible and Orchard Hierarchy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/af0qj3z860hgc",
  "venue": "arXiv",
  "versions": []
}