[Paper Review] Combinatorial comparison of general galled trees, time-consistent galled trees, and simplex time-consistent galled trees

Lily Agranat-Tamir, Michael Fuchs+3 more|arXiv (Cornell University)|Jan 12, 2026

Genomics and Phylogenetic Studies0 citations

TL;DR

The paper compares three classes of rooted binary galled trees (general, time-consistent, and simplex time-consistent) and derives their unlabeled and leaf-labeled enumerative counts, showing that with a fixed number of galls they are asymptotically equivalent, while without fixing galls they diverge.

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.

Motivation & Objective

• Motivate and extend enumeration of leaf-labeled and unlabeled rooted binary phylogenetic networks with galls.
• Determine whether general galled trees and simplex time-consistent galled trees share the same asymptotics as time-consistent galled trees for a fixed number of galls.
• Provide recursions and generating function frameworks to count small-leaf instances and to derive asymptotic behavior across classes.

Proposed method

• Develop unlabeled and leaf-labeled generating functions for general galled trees, time-consistent galled trees, and simplex time-consistent galled trees.
• Use symbolic method and bivariate generating functions to encode leaves and galls, including derivatives to fix the number of galls.
• Derive asymptotic approximations as the number of leaves grows, with and without fixing the number of galls (Sections 3–6).
• Compare asymptotic growth across classes via extracted coefficients from the generating functions and use Faà di Bruno and Leibniz rules for derivatives (Sections 3.5–3.6, 4.5–4.6).
• Obtain recurrences to compute exact counts for small leaves (Section 7).

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

Experimental results

Research questions

• RQ1 Do general galled trees and time-consistent galled trees share the same asymptotic counts when the number of galls is fixed?
• RQ2 Does the simplex time-consistent class have the same fixed-gall asymptotics as the other time-consistent families?
• RQ3 How do the asymptotics differ among the three classes when the number of galls is not restricted?
• RQ4 What are the exact counting recurrences for small leaf counts across the classes?
• RQ5 What are the differences between labeled and unlabeled enumerations for these classes?

Key findings

• For a fixed number of galls, the asymptotic count of general galled trees matches that of time-consistent galled trees as the number of leaves grows; simplex time-consistent galled trees have a smaller asymptotic count.
• When the number of galls is not fixed, simplex time-consistent galled trees are fewer in number than time-consistent galled trees, which are fewer than general galled trees.
• The paper provides recursions to count networks with small numbers of leaves and fixed galls, plus enumerative results for unlabeled networks in the investigated classes.
• Unlabeled and leaf-labeled counts are derived within each class, and the aysmptotic behavior is shown to align with the time-consistent baseline for fixed galls while diverging when galls are unrestricted.
• The analysis confirms that general galled trees with a fixed number of galls behave asymptotically like time-consistent galled trees with a fixed number of galls, but simplex time-consistent galled trees behave differently under the fixed-gall constraint.

Figure 2: The three structures possible for a (non-plane) unlabeled general galled tree with exactly one gall.