On The Exact Counting of Tree-Child Networks

Abstract

The combinatorial study of phylogenetic networks has attracted much attention in recent times. In particular, one class of them, the so-called tree-child networks, are becoming the most prominent ones. However, their combinatorial properties are largely unknown. In this paper we address the problem of exactly counting them. We conjecture a bijection with a certain class of words, and from this assumption a simple recurrence formula arises. It is able to determine the number of all subclasses, as well as a direct formula, a simple enumeration procedure and precise asympotics. Our results coincide with all currently proved formulas for particular subclasses of tree-child networks, as well as with numerical results obtained for small networks. Since, as we will show, working with words greatly simplies the problem, we expect to contribute to further combinatoric characterizations of this class of networks.

A Short Note on the Exact Counting of Tree-Child Networks

“…This conjecture is consistent with previous results about the counting of tree-child networks and has several interesting consequences; see [4]. The purpose of this short note is to verify the above conjecture for one-component tree-child networks which are an important building block of (general) tree-child networks; see [1].…”

“…We give a consequence of Theorem 1 and compare our map with the map from [4] (k = 0) and the restriction of the map of [2] (k = n − 1) to OTCs.…”

“…We next discuss how our bijective map compares to the ones from [4] and [2] for k = 0 and k = n − 1, respectively. First, the one from [4] for k = 0 coincides with our map.…”

“…One of the most popular of these classes is the class of tree-child networks. Consequently, several recent studies have investigated the properties of this class; see [1,2,4] and references therein.…”

“…Several recent papers have investigated counting questions for tree-child networks; see [1,2,4]. One fruitful observation was that tree-child networks can be encoded by words.…”