{
  "id": "2308.06580",
  "version": "v1",
  "published": "2023-08-12T14:35:37.000Z",
  "updated": "2023-08-12T14:35:37.000Z",
  "title": "Universal rooted phylogenetic tree shapes and universal tanglegrams",
  "authors": [
    "Ann Clifton",
    "Eva Czabarka",
    "Kevin Liu",
    "Sarah Loeb",
    "Utku Okur",
    "Laszlo Szekely",
    "Kristina Wicke"
  ],
  "comment": "18 pages, 14 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We provide an $\\Omega(n\\log n) $ lower bound and an $O(n^2)$ upper bound for the smallest size of rooted binary trees (a.k.a. phylogenetic tree shapes), which are universal for rooted binary trees with $n$ leaves, i.e., contain all of them as induced binary subtrees. We explicitly compute the smallest universal trees for $n\\leq 11$. We also provide an $\\Omega(n^2) $ lower bound and an $O(n^4)$ upper bound for the smallest size of tanglegrams, which are universal for size $n$ tanglegrams, i.e., which contain all of them as induced subtanglegrams. Some of our results generalize to rooted $d$-ary trees and to $d$-ary tanglegrams.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-12T14:35:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C35",
      "05C60"
    ],
    "keywords": [
      "universal rooted phylogenetic tree shapes",
      "universal tanglegrams",
      "rooted binary trees",
      "upper bound",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}