{
  "abstract": "We determine the asymptotic height of the discrete-time critical beta-splitting tree. Let $L_n^*$ denotes the height of the tree with $n$ leaves, defined as the maximum graph distance from the root to a leaf. Then, \\[ \\frac{L_n^*}{(\\log n)^2} \\longrightarrow C_{\\mathrm{ht}}:=\\min_{θ>1} \\fracθ{2\\{ψ(θ)+γ\\}} \\approx 0.976 \\] almost surely and in $L^{p}$ for every fixed $p>1$ as $n \\to \\infty$. Here $ψ$ is the digamma function and $γ$ is Euler's constant. This answers \\cite[Open Problem~4]{AldousJansonII} of Aldous and Janson.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14329",
  "author_total": 1,
  "authors": [
    {
      "name": "Heng Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c4v93w0yv8sap.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14329",
  "fulltext_url": null,
  "id": "c4v93w0yv8sap",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14329",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213271606",
  "pdf_url": "https://arxiv.org/pdf/2609.14329",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c4v93w0yv8sap.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.14329",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14329"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The height of discrete-time critical beta-splitting trees",
  "title_text": "The height of discrete-time critical beta-splitting trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/c4v93w0yv8sap",
  "venue": "arXiv",
  "versions": []
}