{
  "id": "1207.3915",
  "version": "v2",
  "published": "2012-07-17T08:44:04.000Z",
  "updated": "2013-05-20T03:47:36.000Z",
  "title": "The asymptotic number of different rooted trees of a tree",
  "authors": [
    "Xueliang Li",
    "Yiyang Li",
    "Yongtang Shi"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{T}_n$ be the set of trees with $n$ vertices. Suppose that each tree in $\\mathcal{T}_n$ is equally likely. We show that the number of different rooted trees of a tree equals $(\\mu_r+o(1))n$ for almost every tree of $\\mathcal{T}_n$, where $\\mu_r$ is a constant. As an application, we show that the number of any given pattern in $\\mathcal{T}_n$ is also asymptotically normally distributed with mean $\\sim \\mu_M n$ and variance $\\sim \\sigma_M n$, where $\\mu_M, \\sigma_M$ are some constants related to the given pattern. This solves an open question claimed in Kok's thesis.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-20T03:47:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C30",
      "05D40",
      "05A15",
      "05A16"
    ],
    "keywords": [
      "rooted trees",
      "asymptotic number",
      "koks thesis",
      "tree equals",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.3915L"
    }
  }
}