{
  "abstract": "Let $\\cP_{\\ell,k}$ be the set of rooted binary phylogenetic networks with $\\ell$ labelled leaves, $k$ reticulations and no parallel edges. We write $|\\cP_{\\ell,k}|=W_k(\\ell)+D_k(\\ell)$, where the weighted count $W_k(\\ell)$ adds the inverse orders of the leaf-fixing automorphism groups and the defect $D_k(\\ell)$ collects the remainder. The weighted count satisfies, for every $k$, a recursion over the source layers of the tree-component structure that involves neither a list of component graphs nor any distinction between symmetric and asymmetric configurations, and its exponential generating function is a Laurent polynomial in $\\sqrt{1-2x}$. Automorphism groups of networks are $2$-groups, elementary abelian for $k\\le5$ but not in general. For $k\\le5$ the defect is the weighted count of networks with a distinguished involution, which obeys an extension of the same recursion. An exact symbolic evaluation of the two recursions yields $|\\cP_{\\ell,k}|$ in closed form for $k\\le5$, the case $k=5$ being new; it reproduces the published counts for $k\\le4$ and corrects a coefficient in a published generating function for $k=3$. For every $k$, uniformly over explicit ranges of $k$, we prove that the non-tree-child networks are a fraction $2k(k-1)/\\ell$ of the tree-child networks to leading order, which gives the third term of the asymptotic expansion of $|\\cP_{\\ell,k}|$. We also prove that reticulation-visible networks exceed tree-child networks by the fraction $k(k-1)/\\ell$, and that a uniformly random network in $\\cP_{\\ell,k}$ has a nontrivial automorphism with probability $k(k-1)/(4\\ell^3)$ to leading order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03856",
  "author_total": 1,
  "authors": [
    {
      "name": "Josep Batle"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1h96w9thw61qw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03856",
  "fulltext_url": null,
  "id": "1h96w9thw61qw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03856",
  "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": "W7220748023",
  "pdf_url": "https://arxiv.org/pdf/2610.03856",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1h96w9thw61qw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03856",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03856"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.PE",
      "subfield": "evolutionary-biology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact and asymptotic enumeration of unrestricted binary phylogenetic networks through automorphism weights",
  "title_text": "Exact and asymptotic enumeration of unrestricted binary phylogenetic networks through automorphism weights",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:11Z",
  "url": "https://research.draconitic.com/paper/1h96w9thw61qw",
  "venue": "arXiv",
  "versions": []
}