On the asymptotic growth of the number of tree-child networks

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

doi:10.1016/j.ejc.2020.103278

Cited by 22 publications

(55 citation statements)

References 4 publications

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“…Specifically for k = 1, 2 and 3, see Refs. [23,5,11] respectively, and for TCNs with the maximum number of reticulation nodes [13]. Our conjectured formula, Eq.…”

Section: Introductionmentioning

confidence: 86%

“…The first considered case is the maximally reticulated TCN. The mapping, to words having every letter repeated exactly three times, was given by Fuchs et al [13]. The second case is the network without reticulations, that is, a tree.…”

Section: Tree-child Network and Wordsmentioning

confidence: 99%

“…Thus, the resulting map constitutes a bijection. Fuchs et al [13] made use of these words as an auxiliary tool to determine the cardinality of the |T C n,n−1 | subclass, and to discern its asymptotic behavior. However, they are important in their own right because statistical properties of the words have a direct reflection on the topology of the network.…”

Section: Maximally Reticulated Tree-child Networkmentioning

confidence: 99%

“…Due to the increasing popularity of the usage of phylogenetic networks, a combinatorial approach to their study regarding counting, enumeration and stochastic characterization has brought much attention recently [5,19,10,11,12,13,23,3,15,1]. It is usual to impose further restrictions to the general structure of phylogenetic networks (in general they are labeled directed acyclic graphs) in order to make them more manageable.…”

Section: Introductionmentioning

confidence: 99%

“…In section III we exhibit the parallelism between TCNs and words. We start by reproducing the bijection given by Fuchs et al [13] for the maximally reticulated subclass. In the second part of the section we establish a bijection between phylogenetic trees and a class of very similar words.…”

Section: Introductionmentioning

confidence: 99%

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On The Exact Counting of Tree-Child Networks

Pons

Batle

2021

Preprint

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The combinatorial study of phylogenetic networks has attracted much attention in recent times. In particular, one class of them, the so-called tree-child networks, are becoming the most prominent ones. However, their combinatorial properties are largely unknown. In this paper we address the problem of exactly counting them. We conjecture a bijection with a certain class of words, and from this assumption a simple recurrence formula arises. It is able to determine the number of all subclasses, as well as a direct formula, a simple enumeration procedure and precise asympotics. Our results coincide with all currently proved formulas for particular subclasses of tree-child networks, as well as with numerical results obtained for small networks. Since, as we will show, working with words greatly simplies the problem, we expect to contribute to further combinatoric characterizations of this class of networks.

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“…Specifically for k = 1, 2 and 3, see Refs. [23,5,11] respectively, and for TCNs with the maximum number of reticulation nodes [13]. Our conjectured formula, Eq.…”

Section: Introductionmentioning

confidence: 86%

Section: Tree-child Network and Wordsmentioning

confidence: 99%

Section: Maximally Reticulated Tree-child Networkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On The Exact Counting of Tree-Child Networks

Pons

Batle

2021

Preprint

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Combinatorial and stochastic properties of ranked tree‐child networks

Bienvenu

Lambert

Steel

2021

Random Struct Algorithms

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Tree-child networks are a class of directed acyclic graphs that have recently risen to prominence in phylogenetics. Although these networks have numerous, attractive mathematical properties, many combinatorial questions concerning them remain intractable. We show that endowing tree-child networks with a biologically relevant ranking structure yields mathematically tractable objects, which we term ranked tree-child networks (RTCNs). We derive explicit enumerative formulas and explain how to sample RTCNS uniformly at random. We study the properties of uniform RTCNs, including: lengths of random walks between root and leaves; distribution of number of cherries in the network; and sampling RTCNs conditional on displaying a given tree. We also formulate a conjecture regarding the scaling limit of the process counting the number of lineages in the ancestry of a leaf. The main idea in this paper, namely using ranking as a way to achieve combinatorial tractability, may also extend to other classes of networks.

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Limit theorems for patterns in ranked tree‐child networks

Fuchs

Liu

2023

Random Struct Algorithms

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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. (Random Struct. Algoritm. 60 (2022), no. 4, 653–689). 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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On the asymptotic growth of the number of tree-child networks

Cited by 22 publications

References 4 publications

On The Exact Counting of Tree-Child Networks

On The Exact Counting of Tree-Child Networks

Combinatorial and stochastic properties of ranked tree‐child networks

Limit theorems for patterns in ranked tree‐child networks

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