All galls are divided into three or more parts: recursive enumeration of labeled histories for galled trees

Abstract

Objective In mathematical phylogenetics, a labeled rooted binary tree topology can possess any of a number of labeled histories, each of which represents a possible temporal ordering of its coalescences. Labeled histories appear frequently in calculations that describe the combinatorics of phylogenetic trees. Here, we generalize the concept of labeled histories from rooted phylogenetic trees to rooted phylogenetic networks, specifically for the class of rooted phylogenetic networks known as rooted galled trees. Results Extending a recursive algorithm for enumerating the labeled histories of a labeled tree topology, we present a method to enumerate the labeled histories associated with a labeled rooted galled tree. The method relies on a recursive decomposition by which each gall in a galled tree possesses three or more descendant subtrees. We exhaustively provide the numbers of labeled histories for all small galled trees, finding that each gall reduces the number of labeled histories relative to a specified galled tree that does not contain it. Conclusion The results expand the set of structures for which labeled histories can be enumerated, extending a well-known calculation for phylogenetic trees to a class of phylogenetic networks.

Enumeration of Rooted Binary Unlabeled Galled Trees

“…Chang et al. ( 2018 ) and Mathur and Rosenberg ( 2023 ) have posed the problem of enumerating rooted binary galled trees in which the leaves are not labeled. In a study focused on introducing encodings for galled trees, Chang et al.…”

“…( 2018 ) argued that the number of rooted binary unlabeled galled trees with n leaves is bounded above by a sequence with a certain generating function. In an enumerative study of labeled histories for rooted binary leaf-labeled galled trees, Mathur and Rosenberg ( 2023 ) enumerated a class of rooted binary unlabeled galled trees for n from 1 to 6, obtaining 1, 1, 2, 6, 20, 72. These values are indeed bounded above by the corresponding upper bounds of Chang et al.…”

“…How many rooted binary unlabeled galled trees possess a given number of leaves n and a given number of galls g ? Here, we perform this general enumeration, counting the rooted binary unlabeled galled trees of Mathur and Rosenberg ( 2023 ) for leaves and galls. We first recursively enumerate all such rooted binary unlabeled galled trees with a specified number of leaves n , considering all possible numbers of galls.…”

“…We follow Mathur and Rosenberg ( 2023 ) in describing key concepts, assuming that all networks and trees are binary (and henceforth dropping the term binary ). A rooted phylogenetic network is a directed acyclic graph with four properties: (i) there exists a unique root node with in-degree 0 and out-degree 2; (ii) all leaf nodes have in-degree 1 and out-degree 0; (iii) all non-leaf, non-root nodes have in-degree 2 and out-degree 1 or in-degree 1 and out-degree 2; and (iv) all edges are directed away from the root.…”

“…First, (i) each reticulation node has a unique ancestor node r such that exactly two non-overlapping paths of edges exist from r to ; if the direction of edges is ignored, then the two paths connecting r and form a cycle , known as a gall . Following Mathur and Rosenberg ( 2023 ), the ancestor node r must be separated from by at least two edges. This requirement that cycles contain at least four nodes is required by the perspective of Mathur and Rosenberg ( 2023 ) that views galled trees as evolving temporally by a biological process such as hybridization.…”