{
  "abstract": "We study the sizes of optimal packings and coverings of conditioned branching processes by balls with growing radii. We find a universal limiting constant, and thereby establish universality of packing and covering numbers across different scales and different offspring distributions. To be precise, for $μ$ an offspring distribution with mean $1$ and finite variance $σ^2$ we let $T_n$ be a Bienaymé tree with offspring distribution $μ$ conditioned to have size $n$. We prove that the packing number (resp. covering number) of $T_n$ with balls of radii $1\\ll γ_n \\ll n^{1/2}$ is asymptotically equivalent to $\\mathfrak{C} n /(σ^2 γ_n^2)$ (resp. $\\mathfrak{C} n /(4σ^2 γ_n^2)$), for a universal constant $\\mathfrak{C}$. We also show that the order of limits can be exchanged and that the same limit occurs when we first consider the limiting covering number of $T_n$ by balls of fixed radius $R$, and thereafter let $R$ tend to $\\infty$. We then extend these results to the Brownian continuum random tree conditioned to have mass $1$. For this model, we prove that the packing (resp. covering) number with balls of radii $\\varepsilon$ are asymptotically equivalent to $\\mathfrak{C}/\\varepsilon^2$ (resp. $\\mathfrak{C}/(4\\varepsilon^2)$), for the same constant $\\mathfrak{C}$, in contrast with a conjecture by Aldous. Finally, our results answer a question by Devroye, Eide and Pralat on the asymptotic burning number of mean $1$ finite variance Bienaymé trees of size $n$, by showing that it lies between $(c+o(1)) n^{1/3}$ and $(2c+o(1))n^{1/3}$ with high probability, for a constant $c$ that only depends on the offspring variance.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40150",
  "author_total": 4,
  "authors": [
    {
      "name": "Caelan Atamanchuk"
    },
    {
      "name": "Simon Briend"
    },
    {
      "name": "Serte Donderwinkel"
    },
    {
      "name": "Elaine Herrera"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4gskm57mkreyz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40150",
  "fulltext_url": null,
  "id": "4gskm57mkreyz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40150",
  "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",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40150",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4gskm57mkreyz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40150",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40150"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Packing and covering random trees",
  "title_text": "Packing and covering random trees",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/4gskm57mkreyz",
  "venue": "arXiv",
  "versions": []
}