Counting and enumerating galled networks

Abstract

Galled trees are widely studied as a recombination model in population genetics. This class of phylogenetic networks is generalized into galled networks by relaxing a structural condition. In this work, a linear recurrence formula is given for counting 1galled networks, which are galled networks satisfying the condition that each reticulate node has only one leaf descendant. Since every galled network consists of a set of 1-galled networks stacked one on top of the other, a method is also presented to count and enumerate galled networks.

On the Asymptotic Growth of the Number of Tree-Child Networks

“…because the third ω n has exactly 3n − 2 − i positions to choose from. Finally, by setting b n,i := a n,n−i we obtain the claimed recurrence from ( 6) and (7) with the claimed initial conditions (which are easily verified). Using this recurrence, the first, e.g., 1000 terms of a n can be computed with Maple in a few seconds.…”

“…Over the last two decades, phylogenetic networks have become increasingly popular and have been used more and more frequently in modeling horizontal genetic transfer events in evolutionary genomics. Because of their now widespread usage, studying basic combinatorial properties such as counting them has attracted some recent efforts; see, e.g., Bouvel et al [1], Cardona and Zhang [2], Fuchs et al [5], Gunawan et al [7], McDiarmid at al. [8], and Zhang [10].…”

On The Exact Counting of Tree-Child Networks

“…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.…”

The Sackin Index of Simplex Networks

“…Counting ranked tree-child networks is studied in [14]. In addition, asymptotic and exact counts of galled trees and galled networks are given in [15] and [16,17], respectively.…”

“…Eqn (16),. we have that2K(OC n ) ≥ D C (n) = 2( √ 2 − 1) √ πn 7/4 + O(1), equivalently, K(OC n ) ≥ ( √ 2 − 1) √ πn 7/4 + O(1).…”