{
  "abstract": "We investigate scaling and local limits of random trees biased according to their number of descents. A descent in a rooted labeled tree $t$ is a parent-child pair such that the label of the parent is greater than the label of the child, and the total number of descents is denoted by $\\mathrm{des}(t)$. For $n \\geq 1$ and $q_n \\geq 0$, we consider the probability measure on trees of size $n$ where each tree $t$ is chosen with a probability proportional to $q_n^{\\mathrm{des}(t)}$. We study the resulting random tree $\\mathcal{T}_n^{(q_n)}$ properly rescaled as $n \\to \\infty$, and focus on two regimes for the bias parameter $q_n$. When $q_n = q \\in (0,1]$ is fixed, we prove that $\\mathcal{T}_n^{(q)}$ converges in distribution to the Brownian Continuum Random Tree. When $q_n = a/n$ for some fixed $a > 0$, we prove that $\\mathcal{T}_n^{(q_n)}$ converges in distribution to a random non-trivial dendron constructed from a Poisson-Dirichlet sequence. We complement these results with a description of the Benjamini-Schramm local limit of $\\mathcal{T}_n^{(q_n)}$ in all regimes of parameters. Our proofs rely on analytic combinatorics to find the asymptotics of certain statistics of the tree, and on a probabilistic analysis of the structure of descent-biased random trees.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15451",
  "author_total": 3,
  "authors": [
    {
      "name": "Victor Dubach"
    },
    {
      "name": "Paul Thévenin"
    },
    {
      "name": "Stephan Wagner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45n83zjhexdph.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15451",
  "fulltext_url": null,
  "id": "45n83zjhexdph",
  "issns": [],
  "landing_url": "https://hal.science/hal-05754228v1",
  "language": "en",
  "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",
      "math.PR"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213255661",
  "pdf_url": "https://arxiv.org/pdf/2609.15451",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45n83zjhexdph.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:36Z",
      "native_id": "hal-05754228",
      "source": "hal",
      "url": "https://hal.science/hal-05754228v1"
    },
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15451",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15451"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Limits of descent-biased trees",
  "title_text": "Limits of descent-biased trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/45n83zjhexdph",
  "venue": "HAL",
  "versions": []
}