Counting phylogenetic networks with few reticulation vertices: A second approach

Fuchs, Michael; Huang, En-Yu; Yu, Guan-Ru

doi:10.1016/j.dam.2022.03.026

Cited by 7 publications

(4 citation statements)

References 18 publications

(30 reference statements)

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“…A similar moderation in growth with increasing n is just observable for , which could hint at a similar exponential growth; we can conjecture that each with fixed g has the same exponential growth. Note that Fuchs et al ( 2019 , Theorem 5.1; 2022 , Theorem 1 and Corollary 2) showed that the exponential growth of labeled tree-child networks and normal networks with a fixed number of reticulation vertices (corresponding to a fixed number of galls in our case) is the same for any such number; only the subexponential growth differs.…”

Section: Discussionmentioning

confidence: 68%

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Enumeration of Rooted Binary Unlabeled Galled Trees

Agranat-Tamir,

Mathur,

Rosenberg

2024

Bull Math Biol

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Rooted binary galled trees generalize rooted binary trees to allow a restricted class of cycles, known as galls. We build upon the Wedderburn-Etherington enumeration of rooted binary unlabeled trees with n leaves to enumerate rooted binary unlabeled galled trees with n leaves, also enumerating rooted binary unlabeled galled trees with n leaves and g galls, $$0 \leqslant g \leqslant \lfloor \frac{n-1}{2} \rfloor $$ 0 ⩽ g ⩽ ⌊ n - 1 2 ⌋ . The enumerations rely on a recursive decomposition that considers subtrees descended from the nodes of a gall, adopting a restriction on galls that amounts to considering only the rooted binary normal unlabeled galled trees in our enumeration. We write an implicit expression for the generating function encoding the numbers of trees for all n. We show that the number of rooted binary unlabeled galled trees grows with $$0.0779(4.8230^n)n^{-\frac{3}{2}}$$ 0.0779 ( 4 . 8230 n ) n - 3 2 , exceeding the growth $$0.3188(2.4833^n)n^{-\frac{3}{2}}$$ 0.3188 ( 2 . 4833 n ) n - 3 2 of the number of rooted binary unlabeled trees without galls. However, the growth of the number of galled trees with only one gall has the same exponential order 2.4833 as the number with no galls, exceeding it only in the subexponential term, $$0.3910n^{\frac{1}{2}}$$ 0.3910 n 1 2 compared to $$0.3188n^{-\frac{3}{2}}$$ 0.3188 n - 3 2 . For a fixed number of leaves n, the number of galls g that produces the largest number of rooted binary unlabeled galled trees lies intermediate between the minimum of $$g=0$$ g = 0 and the maximum of $$g=\lfloor \frac{n-1}{2} \rfloor $$ g = ⌊ n - 1 2 ⌋ . We discuss implications in mathematical phylogenetics.

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Section: Discussionmentioning

confidence: 68%

“…( 2022 ), Fuchs et al. ( 2022 )] and is one of relatively few studies to examine a class of unlabeled networks (Chang et al. 2018 ; Mathur and Rosenberg 2023 ).…”

Section: Discussionmentioning

confidence: 99%

Enumeration of Rooted Binary Unlabeled Galled Trees

Agranat-Tamir,

Mathur,

Rosenberg

2024

Bull Math Biol

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“…The situation for tree-child networks is slightly better, since there are previous works on the enumeration (Fuchs et al. 2021 ; Pons and Batle 2021 ) and generation (Cardona et al. 2019 ; Cardona and Zhang 2020 ) of this kind of networks, but much less efficiently than the method given here (see Sect.…”

Section: Introductionmentioning

confidence: 82%

Generation of Orchard and Tree-Child Networks

Cardona,

Ribas,

Pons

2023

Bull Math Biol

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Phylogenetic networks are an extension of phylogenetic trees that allow for the representation of reticulate evolution events. One of the classes of networks that has gained the attention of the scientific community over the last years is the class of orchard networks, that generalizes tree-child networks, one of the most studied classes of networks. In this paper we focus on the combinatorial and algorithmic problem of the generation of binary orchard networks, and also of binary tree-child networks. To this end, we use that these networks are defined as those that can be recovered by reversing a certain reduction process. Then, we show how to choose a “minimum” reduction process among all that can be applied to a network, and hence we get a unique representation of the network that, in fact, can be given in terms of sequences of pairs of integers, whose length is related to the number of leaves and reticulations of the network. Therefore, the generation of networks is reduced to the generation of such sequences of pairs. Our main result is a recursive method for the efficient generation of all minimum sequences, and hence of all orchard (or tree-child) networks with a given number of leaves and reticulations. An implementation in C of the algorithms described in this paper, along with some computational experiments, can be downloaded from the public repository https://github.com/gerardet46/OrchardGenerator. Using this implementation, we have computed the number of binary orchard networks with at most 6 leaves and 8 reticulations.

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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][16][17]24] and references given therein. In the present work, we introduce branching process methods to their study.…”

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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Counting phylogenetic networks with few reticulation vertices: A second approach

Cited by 7 publications

References 18 publications

Enumeration of Rooted Binary Unlabeled Galled Trees

Enumeration of Rooted Binary Unlabeled Galled Trees

Generation of Orchard and Tree-Child Networks

A branching process approach to level‐k phylogenetic networks

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