Generating normal networks via leaf insertion and nearest neighbor interchange

Abstract

BackgroundGalled trees are studied as a recombination model in theoretical population genetics. This class of phylogenetic networks has been generalized to tree-child networks and other network classes by relaxing a structural condition imposed on galled trees. Although these networks are simple, their topological structures have yet to be fully understood.ResultsIt is well-known that all phylogenetic trees on n taxa can be generated by the insertion of the n-th taxa to each edge of all the phylogenetic trees on n−1 taxa. We prove that all tree-child (resp. normal) networks with k reticulate nodes on n taxa can be uniquely generated via three operations from all the tree-child (resp. normal) networks with k−1 or k reticulate nodes on n−1 taxa. Applying this result to counting rooted phylogenetic networks, we show that there are exactly binary phylogenetic networks with one reticulate node on n taxa.ConclusionsThe work makes two contributions to understand normal networks. One is a generalization of an enumeration procedure for phylogenetic trees into one for normal networks. Another is simple formulas for counting normal networks and phylogenetic networks that have only one reticulate node.

On The Exact Counting of Tree-Child Networks

“…Our conjectured formula, Eq. (9), coincides with all these results, and it also reproduces particular values obtained by demanding computational procedures [23,5].…”

“…Furthermore simple analytic expressions can be extracted from the hypothesis, which coincide with already proven formulas for low numbers of reticulation nodes deduced by different methods: case k = 1 solved by L. Zhang in Ref. [23], case k = 2 proved by Cardona and Zhang [5] and k = 3 provided by Fuchs et al [11]. Of course it also coincides with the extreme cases k = 0 and k = n − 1.…”

“…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.…”

“…Last expression, k = 4, is completely new, whereas the previous ones agree and simplify the existing formulas in a compact manner. An equivalent expression to (15a) was first provided by L. Zhang [23], a direct formula for TCNs with two reticulation nodes was first given by Cardona and Zhang [5], while a closed formula for |T C n,3 | can be found in Ref. [11].…”

“…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.…”