{
  "id": "2205.06625",
  "version": "v1",
  "published": "2022-05-13T13:19:44.000Z",
  "updated": "2022-05-13T13:19:44.000Z",
  "title": "The probability of random trees being isomorphic",
  "authors": [
    "Christoffer Olsson"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We show that the probability that two randomly chosen trees are isomorphic decays exponentially for rooted labelled trees as well as Galton--Watson trees with bounded degrees. In the former case a full asymptotic expansion is derived. We also show that, in general, we cannot obtain exponential decay for Galton--Watson trees. Lastly, we prove joint convergence to a multivariate normal distribution for vertices of given degrees in pairs of labelled trees conditioned on being isomorphic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-13T13:19:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05"
    ],
    "keywords": [
      "random trees",
      "probability",
      "galton-watson trees",
      "full asymptotic expansion",
      "multivariate normal distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}