Enumerative combinatorics of unlabeled and labeled time-consistent galled trees

TL;DR

The paper addresses the enumerative problem for time-consistent galled trees, a constrained class of rooted binary phylogenetic networks, by deriving generating functions for both unlabeled and leaf-labeled cases using the symbolic method. It delivers explicit generating-function formulas for no galls, one gall, two galls, and arbitrary numbers of galls, including bivariate and fixed-$g$ variants, and obtains asymptotic growth rates that enable comparison with other network classes. In the unlabeled setting, closed forms and recursions are established for counts by leaves and galls, while in the labeled setting, exponential generating functions yield precise asymptotics such as $e_{n,g}\sim\dfrac{2^{2g-1}\sqrt{2}}{(2g)!}\big(\frac{2}{e}\big)^n n^{n+2g-1}$, with a related factor $1/(2k-1)!!$ reducing counts relative to more permissive classes. Overall, the work clarifies how time-consistency constraints reduce combinatorial complexity compared to broader phylogenetic-network classes and demonstrates the efficacy of the symbolic method for constrained network enumerations, thereby enriching the understanding of the enumerative landscape in phylogenetics.

Abstract

In mathematical phylogenetics, the time-consistent galled trees provide a simple class of rooted binary network structures that can be used to represent a variety of different biological phenomena. We study the enumerative combinatorics of unlabeled and labeled time-consistent galled trees. We present a new derivation via the symbolic method of the number of unlabeled time-consistent galled trees with a fixed number of leaves and a fixed number of galls. We also derive new generating functions and asymptotics for labeled time-consistent galled trees.

Figures (5)

• Figure 1: Examples of phylogenetic networks. Sources for definitions appear in Table \ref{['table:definitions']}. (A) Time-consistent galled tree (equivalently, normal galled tree). We draw reticulation events on a horizontal line to represent the concurrent existence of two merging entities that produce a hybrid entity. (B) Galled tree. This network is not a time-consistent galled tree because it has a reticulation cycle in which the two parents of the reticulation are a parent--child pair themselves (blue). (C) Galled tree-child network. This network is not a galled tree because it has nodes that are part of more than one reticulation cycle (blue). (D) Normal network. This network is not a normal (time-consistent) galled tree because it has nodes that are part of more than one reticulation cycle (blue). (E) Tree-child network. This network is not a galled tree-child network because it has a reticulation node that is in two reticulation cycles (e.g. blue). It is not a normal network because it contains a "shortcut" (red). (F) Galled network. This network is not a galled tree-child network because it has a tree node that has only reticulation nodes as children (blue). (G) Reticulation-visible network. This network is not a galled network because it has a reticulation node that is in two reticulation cycles (red). It is not a tree-child network because it has a tree node that is a parent of two reticulation nodes (blue). (H) Phylogenetic network. This network is not a reticulation-visible network because it has a reticulation node all of whose descendant leaves possess paths from the root that do not traverse it (blue).
• Figure 2: Inclusion relations between classes of phylogenetic networks. Arrows represent the inclusion of the class on the top in the class below it. Notice that time-consistent galled trees are also normal galled trees because time-consistency implies no "shortcuts" (in a galled tree, a shortcut can only appear in a gall, contradicting time-consistency). Inclusion relationships, indicated by numbers, are described in Table \ref{['table:inclusion']}.
• Figure 3: A non-plane unlabeled time-consistent galled tree with one gall, $\mathcal{E}_1$, has one of two structures. (A) A root with one subtree with no galls ($\mathcal{U}$) and one subtree with one gall ($\mathcal{E}_1$); there is no case of symmetry in this scenario. (B) A root gall, which is the only gall, and a subtree with no galls descended from the reticulation node (bold), with two non-empty sequences of subtrees with no galls (non-empty because of the time-consistency condition). The latter scenario has a case of symmetry, and because the trees are unlabeled in addition to being non-plane, the two non-empty sequences form a multiset of size 2.
• Figure 4: A non-plane unlabeled time-consistent galled tree with two galls, $\mathcal{E}_2$, has one of four structures. (A) A root with one subtree with no galls ($\mathcal{U}$) and one subtree with two galls ($\mathcal{E}_2$); there is no case of symmetry. (B) A root with two subtrees each with one gall ($\mathcal{E}_1$). This scenario has a case of symmetry, and because the tree is both non-plane and unlabeled, the two subtrees form a multiset of size 2. (C) A root gall and a subtree with one gall descended from the reticulation node (bold). On both sides of the reticulation node, because of the time-consistency condition, there are non-empty sequences of subtrees with no galls; because of symmetry, the two non-empty sequences form a multiset of size 2. (D) A root gall and a subtree with no galls descended from the reticulation node (bold). On one side of the reticulation node, there is a non-empty sequence of subtrees with no galls. On the other side, there is a subtree with one gall (to complete the tally of two galls), before and after which are are two possibly empty (because the subtree with the one gall is sufficient for time-consistency) sequences of subtrees with no galls (gray). There is no case of symmetry.
• Figure 5: A non-plane unlabeled time-consistent galled tree with any number of galls, $\mathcal{G}$, has one of three structures. (A) A single leaf, $t$. (B) A root whose two subtrees are both time-consistent non-plane unlabeled galled trees with any number of galls. This scenario has a case of symmetry, and the two subtrees form a multiset of size 2. (C) A root gall, $u$, and a non-plane unlabeled time-consistent galled tree with any number of galls descended from the reticulation node (bold), following two non-empty (because of the time-consistency) sequences of non-plane unlabeled time-consistent galled trees with any number of galls. This scenario has a case of symmetry, and hence, the two non-empty sequences form a multiset of size 2.

Theorems & Definitions (4)

• Proposition 1
• proof
• Proposition 2
• proof