{
  "id": "2010.08613",
  "version": "v1",
  "published": "2020-10-16T20:12:13.000Z",
  "updated": "2020-10-16T20:12:13.000Z",
  "title": "The Horton-Strahler Number of Conditioned Galton-Watson Trees",
  "authors": [
    "Anna M. Brandenberger",
    "Luc Devroye",
    "Tommy Reddad"
  ],
  "comment": "26 pages, 3 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "The Horton-Strahler number of a tree is a measure of its branching complexity; it is also known in the literature as the register function. We show that for critical Galton-Watson trees with finite variance conditioned to be of size $n$, the Horton-Strahler number grows as $\\frac{1}{2}\\log_2 n$ in probability. We further define some generalizations of this number. Among these are the rigid Horton-Strahler number and the $k$-ary register function, for which we prove asymptotic results analogous to the standard case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-16T20:12:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60J80",
      "05C80",
      "05C05"
    ],
    "keywords": [
      "conditioned galton-watson trees",
      "horton-strahler number grows",
      "rigid horton-strahler number",
      "ary register function",
      "standard case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}