{
  "abstract": "We study the extreme local structure of plane binary trees through the limiting distribution of the number of leaves at maximum depth. For every $m\\geq1$, we determine the limiting probability $κ^{[m]}$ that the deepest level contains $2m$ leaves. The resulting sequence is encoded by a generating function $K(y)$, obtained from a bivariate refinement of the Catalan iteration. We show that, at the critical point of this iteration, a parabolic recurrence governs the marked generating functions. Using continuous iteration theory and a Fatou coordinate, we transform this recurrence into the functional equation $K((1+y/4)^2)=K(y)+4$, which describes the limiting law. Its singular behavior yields the exponential tail $κ^{[m]}\\sim4^{-m+1}$. Numerical evaluation gives, e.g., $κ^{[1]}\\approx0.7009$, $κ^{[2]}\\approx0.2226$ and an average number of deepest leaves equal to $\\hatκ=4κ^{[1]}\\approx2.8037$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.12821",
  "author_total": 3,
  "authors": [
    {
      "name": "Olivier Bodini"
    },
    {
      "name": "Antoine Genitrini"
    },
    {
      "name": "Khaydar Nurligareev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/adbd57a31egs6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.12821",
  "fulltext_url": null,
  "id": "adbd57a31egs6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.12821",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7161180351",
  "pdf_url": "https://arxiv.org/pdf/2605.12821",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/adbd57a31egs6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2605.12821",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.12821"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Distribution of the Deepest Leaves in Binary Trees",
  "title_text": "The Distribution of the Deepest Leaves in Binary Trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/adbd57a31egs6",
  "venue": "arXiv",
  "versions": []
}