Asymptotic enumeration and distributional properties of galled networks

Fuchs, Michael; Yu, Guan-Ru; Zhang, Louxin

doi:10.1016/j.jcta.2022.105599

Cited by 7 publications

(10 citation statements)

References 7 publications

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“…With their explicit potential to be temporally ordered, the rooted galled trees here and galled trees in other studies are not generally precisely identical, as the temporal requirement we have imposed is a case of the recent approach of Bienvenu et al [ 1 ] and has not yet been frequently assumed. We have provided an enumeration algorithm for the galled trees we consider; the counts of 1, 1, 2, 6, 20, and 72 for the numbers of rooted unlabeled galled trees for 1 to 6 leaves (Table 6 ) differ from counts and formulas reported in related enumerative studies [ 2 , 4 , 7 , 8 , 21 ]. In studies of labelings for galled trees and phylogenetic networks more generally, care is needed in recognizing the precise set of objects under consideration.…”

Section: Discussionmentioning

confidence: 99%

“…2 Parts of a gall. These include the top node (12), left non-hybridizing side nodes (8), right non-hybridizing side nodes (10,11), left hybridizing side node (6), right hybridizing side node (7), and hybrid node (5). In this galled tree, leaf nodes (orange) are labeled with letters A-K. Internal nodes (black) are numbered using a postorder traversal, with child nodes assigned smaller numbers than parent nodes; at hybridization events, the subtree that receives the smallest numbers is the subtree descended from the hybrid node, and the hybrid node receives a smaller number than the hybridizing side nodes (ii) the two hybridizing nodes that are the parents of the same hybrid node; or (iii) two leaves.…”

Section: Definitionsmentioning

confidence: 99%

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

Mathur

Rosenberg

2023

Algorithms Mol Biol

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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.

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Section: Discussionmentioning

confidence: 99%

Section: Definitionsmentioning

confidence: 99%

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

Mathur

Rosenberg

2023

Algorithms Mol Biol

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“…On the other hand, for phylogenetic networks, which are used to model reticulate evolution, very little is known about the number of occurrences of patterns when the networks from a given class are randomly sampled. This is due to the fact that even counting questions for phylogenetic networks from a given class were still open until recently; see [3,5,16,10,11,12,13,18] for progress on counting questions for some of the major classes of phylogenetic networks.…”

Section: Introductionmentioning

confidence: 99%

Limit Theorems for Patterns in Ranked Tree-Child Networks

Fuchs¹,

Liu²,

Yu³

2022

Preprint

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We prove limit laws for the number of occurrences of a pattern on the fringe of a ranked tree-child network which is picked uniformly at random. Our results extend the limit law for cherries proved by Bienvenu et al. (2022). For patterns of height 1 and 2, we show that they either occur frequently (mean is asymptotically linear and limit law is normal) or sporadically (mean is asymptotically constant and limit law is Poisson) or not all (mean tends to 0 and limit law is degenerate). We expect that these are the only possible limit laws for any fringe pattern.

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“…The application of phylogenetic networks in evolutionary biology motivates the mathematical study of their number and shape, which has received increasing attention in recent literature. See for example [7, 9, 10, 15–17, 24] and references given therein. In the present work, we introduce branching process methods to their study.…”

Section: Introductionmentioning

confidence: 99%

“…Questions concerning heights in various classes of random phylogenetic networks were raised in [24, Sec. 7] and [17, Sec. 8].…”

Section: Introductionmentioning

confidence: 99%

A branching process approach to level‐k phylogenetic networks

Stufler

2021

Random Struct Algorithms

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The mathematical analysis of random phylogenetic networks via analytic and algorithmic methods has received increasing attention in the past years. In the present work we introduce branching process methods to their study. This approach appears to be new in this context. Our main results focus on random level‐k networks with n labeled leaves. Although the number of reticulation vertices in such networks is typically linear in n, we prove that their asymptotic global and local shape is tree‐like in a well‐defined sense. We show that the depth process of vertices in a large network converges towards a Brownian excursion after rescaling by n−1/2. We also establish Benjamini–Schramm convergence of large random level‐k networks towards a novel random infinite network.

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Asymptotic enumeration and distributional properties of galled networks

Cited by 7 publications

References 7 publications

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

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

Limit Theorems for Patterns in Ranked Tree-Child Networks

A branching process approach to level‐k phylogenetic networks

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