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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 . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]