Generating normal networks via leaf insertion and nearest neighbor interchange

• Main Author: Louxin Zhang
• Format: Article
• Language: English
• Published: BMC 2019-12-01
• Series: BMC Bioinformatics
• Subjects: Tree-child networks; normal networks
• Online Access: https://doi.org/10.1186/s12859-019-3209-3

Abstract Background Galled 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. Results It 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 (2n)!2n(n−1)!−2n−1n! $\frac {(2n)!}{2^{n} (n-1)!}-2^{n-1} n!$ binary phylogenetic networks with one reticulate node on n taxa. Conclusions The 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.