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Enumeration of multi-rooted plane trees

Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, Vasilisa Shramchenko

Abstract

We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions.

Enumeration of multi-rooted plane trees

Abstract

We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions.
Paper Structure (8 sections, 10 theorems, 70 equations, 3 figures)

This paper contains 8 sections, 10 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $N$-rooted plane trees
  3. Recursion relations for $N$-rooted trees
  4. Counting $N$-rooted trees
  5. Some explicit examples of our sequences and relations to known sequences
  6. Generating functions for $N$-rooted plane trees
  7. Binomial and hypergeometric identities from $T_N(e, I_N)$
  8. Appendix

Key Result

Lemma 1

For $N$-rooted trees from the set $S_N(e; d_1, \dotsc, d_k)$ with $k\leq N$, the integer quantities $e, \, N, \, d_i$ satisfy Here the vertical bars denote the number of elements in the set. If degrees of $k-1$ of the $N$ root vertices are fixed to be $d_1, \dots, d_{k-1}$, then the highest possible value $D_k$ of the degree $d_k$ of the $k$th root vertex is

Figures (3)

  • Figure 1: The plane tree corresponding to $\alpha=(1\,2)(3\,4)(5\,6)(7\,8)(9\,10)(11\,\,12)(13\,\,14)(15\,\,16)$ and $\sigma=(1)(2\,3)(4\,\,11\,\,9\,7\,5)(6)(8)(10)(12\,\,15\,\,13)(14)(16)$
  • Figure 2: Two 3-rooted trees
  • Figure 3: Five valleys in Dyck paths of length 6

Theorems & Definitions (21)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Remark 2
  • ...and 11 more