Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Combinatorial comparison of general galled trees, time-consistent galled trees, and simplex time-consistent galled trees

Lily Agranat-Tamir, Michael Fuchs, Bernhard Gittenberger, Noah A. Rosenberg, Karthik V. Seetharaman

TL;DR

This work extends the enumerative study of rooted binary phylogenetic networks by analyzing general galled trees and simplex time-consistent galled trees, in both unlabeled and leaf-labeled forms. It shows that, for a fixed number of galls, general galled trees and time-consistent galled trees share the same asymptotic growth, while simplex time-consistent galled trees grow more slowly; when the number of galls is unbounded, the three classes differ in subexponential factors, with simplex TC being smallest. The authors develop a suite of generating functions (univariate, bivariate, and exponential) and recursions to count these networks for small sizes, and derive precise asymptotic formulas, including Catalan- and Wedderburn–Etherington-related constants, as well as explicit asymptotics for maximal-gall cases. The results refine our understanding of the relative abundance of phylogenetic-network classes under gall constraints and provide a foundation for further combinatorial and computational studies in evolutionary modeling.

Abstract

Rooted binary phylogenetic networks are extensions of rooted binary trees, adding reticulation nodes that are designed to represent evolutionary processes that involve hybridization events. Enumerative combinatorics studies have counted leaf-labeled phylogenetic networks in a variety of classes, finding that when the number of reticulations is fixed, the time-consistent galled trees are asymptotically less numerous than each of several network classes that had been previously examined. Here we provide enumerative results on two additional network classes: general galled trees and simplex time-consistent galled trees. We show that for a fixed number of galls, as the number of leaves goes to infinity, the asymptotic count of general galled trees is identical to that of time-consistent galled trees, whereas the count of simplex time-consistent galled trees is smaller. If the number of galls is not restricted, then the asymptotic approximations all differ: simplex time-consistent galled trees are less numerous than time-consistent galled trees, which are in turn less numerous than general galled trees. We also report a variety of additional results: recursions to count the studied networks with small numbers of leaves a fixed number of galls, as well as enumerative results for unlabeled networks in the classes that we investigate.

Combinatorial comparison of general galled trees, time-consistent galled trees, and simplex time-consistent galled trees

TL;DR

This work extends the enumerative study of rooted binary phylogenetic networks by analyzing general galled trees and simplex time-consistent galled trees, in both unlabeled and leaf-labeled forms. It shows that, for a fixed number of galls, general galled trees and time-consistent galled trees share the same asymptotic growth, while simplex time-consistent galled trees grow more slowly; when the number of galls is unbounded, the three classes differ in subexponential factors, with simplex TC being smallest. The authors develop a suite of generating functions (univariate, bivariate, and exponential) and recursions to count these networks for small sizes, and derive precise asymptotic formulas, including Catalan- and Wedderburn–Etherington-related constants, as well as explicit asymptotics for maximal-gall cases. The results refine our understanding of the relative abundance of phylogenetic-network classes under gall constraints and provide a foundation for further combinatorial and computational studies in evolutionary modeling.

Abstract

Rooted binary phylogenetic networks are extensions of rooted binary trees, adding reticulation nodes that are designed to represent evolutionary processes that involve hybridization events. Enumerative combinatorics studies have counted leaf-labeled phylogenetic networks in a variety of classes, finding that when the number of reticulations is fixed, the time-consistent galled trees are asymptotically less numerous than each of several network classes that had been previously examined. Here we provide enumerative results on two additional network classes: general galled trees and simplex time-consistent galled trees. We show that for a fixed number of galls, as the number of leaves goes to infinity, the asymptotic count of general galled trees is identical to that of time-consistent galled trees, whereas the count of simplex time-consistent galled trees is smaller. If the number of galls is not restricted, then the asymptotic approximations all differ: simplex time-consistent galled trees are less numerous than time-consistent galled trees, which are in turn less numerous than general galled trees. We also report a variety of additional results: recursions to count the studied networks with small numbers of leaves a fixed number of galls, as well as enumerative results for unlabeled networks in the classes that we investigate.
Paper Structure (46 sections, 10 theorems, 112 equations, 9 figures, 4 tables)

This paper contains 46 sections, 10 theorems, 112 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. Unlabeled general galled trees
  4. No galls
  5. One gall
  6. Two galls
  7. Bivariate generating function
  8. Fixed number of galls
  9. Arbitrary number of galls
  10. Leaf-labeled general galled trees
  11. No galls
  12. One gall
  13. Two galls
  14. Bivariate exponential generating function
  15. Fixed number of galls
  16. ...and 31 more sections

Key Result

Theorem 1

As $t\rightarrow \rho^-$, the generating function $\mathcal{E}_g^1(t)$ for the number of unlabeled simplex time-consistent galled trees with $g$ galls, $g \geq 1$, satisfies for a constant $\beta_g$. Furthermore, for $g \geq 2$, the constants $\beta_g$ obey the recurrence relation with $\beta_1 = \frac{1}{2}$.

Figures (9)

  • Figure 1: Three classes of phylogenetic networks. (A) A general galled tree. This galled tree is not a time-consistent galled tree because it has a gall (light blue) whose top node is not separated from the reticulation node by at least two edges (it is a parent of the reticulation node). (B) A time-consistent galled tree. This tree is not a simplex galled tree because it has a reticulation node (red) that is the ancestor of a subtree that is not a single leaf (orange). (C) A simplex time-consistent galled tree.
  • Figure 2: The three structures possible for a (non-plane) unlabeled general galled tree with exactly one gall.
  • Figure 3: The six structures possible for a (non-plane) unlabeled general galled tree with exactly two galls.
  • Figure 4: The four structures possible for a (non-plane) unlabeled general galled tree.
  • Figure 5: The two structures possible for a (non-plane) unlabeled simplex time-consistent galled tree with exactly one gall.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Proposition 2
  • Corollary 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Corollary 8
  • Proposition 9
  • Corollary 10