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Phylogenetic networks generalize phylogenetic trees by allowing the modelization of events of reticulate evolution. Among the different kinds of phylogenetic networks that have been proposed in the literature, the subclass of binary tree-child networks is one of the most studied ones. However, very little is known about the combinatorial structure of these networks. In this paper we address the problem of generating all possible binary tree-child (BTC) networks with a given number of leaves in an efficient way via reduction/augmentation operations that extend and generalize analogous operations for phylogenetic trees, and are biologically relevant. Since our solution is recursive, this also provides us with a recurrence relation giving an upper bound on the number of such networks. We also show how the operations introduced in this paper can be employed to extend the evolutive history of a set of sequences, represented by a BTC network, to include a new sequence. An implementation in python of the algorithms described in this paper, along with some computational experiments, can be downloaded from https://github.com/bielcardona/TCGenerators., Research of GC and JCP has been partially supported by the Spanish Ministry of Science, Innovation and Universities (http://www.ciencia.gob.es/) and European Regional Development Fund (https://ec.europa.eu/regional_policy/es/funding/erdf/) projects DPI2015-67082-P and PGC2018-096956-B-C43. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Phylogenetic networks are rooted, directed, acyclic graphs that model reticulate evolutionary histories. Recently, statistical methods were devised for inferring such networks from either gene tree estimates or the sequence alignments of multiple unlinked loci. Bi-allelic markers, most notably single nucleotide polymorphisms (SNPs) and amplified fragment length polymorphisms (AFLPs), provide a powerful source of genome-wide data. In a recent paper, a method called SNAPP was introduced for statistical inference of species trees from unlinked bi-allelic markers. The generative process assumed by the method combined both a model of evolution for the bi-allelic markers, as well as the multispecies coalescent. A novel component of the method was a polynomial-time algorithm for exact computation of the likelihood of a fixed species tree via integration over all possible gene trees for a given marker. Here we report on a method for Bayesian inference of phylogenetic networks from bi-allelic markers. Our method significantly extends the algorithm for exact computation of phylogenetic network likelihood via integration over all possible gene trees. Unlike the case of species trees, the algorithm is no longer polynomial-time on all instances of phylogenetic networks. Furthermore, the method utilizes a reversible-jump MCMC technique to sample the posterior of phylogenetic networks given bi-allelic marker data. Our method has a very good performance in terms of accuracy and robustness as we demonstrate on simulated data, as well as a data set of multiple New Zealand species of the plant genus Ourisia (Plantaginaceae). We implemented the method in the publicly available, open-source PhyloNet software package., Author summary The availability of genomic data has revolutionized the study of evolutionary histories and phylogeny inference. Inferring evolutionary histories from genomic data requires, in most cases, accounting for the fact that different genomic regions could have evolutionary histories that differ from each other as well as from that of the species from which the genomes were sampled. In this paper, we introduce a method for inferring evolutionary histories while accounting for two processes that could give rise to such differences across the genomes, namely incomplete lineage sorting and hybridization. We introduce a novel algorithm for computing the likelihood of phylogenetic networks from bi-allelic genetic markers and use it in a Bayesian inference method. Analyses of synthetic and empirical data sets show a very good performance of the method in terms of the estimates it obtains.