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

Abstract

Tree-child networks are an important network class which are used in phylogenetics to model reticulate evolution. In a recent paper, Pons and Batle (2021) conjectured a relation between tree-child networks and certain words. In this short note, we prove their conjecture for the (important) class of one-component tree-child networks.

Enumeration of $d$-combining Tree-Child Networks

“…Note that the above result for d = 2 is already contained in the proof of Theorem 3 in [8] where even a local limit theorem was proved; see also [7]. Using the above corollary, we now obtain the first order asymptotics of the total number of one-component tree-child networks.…”

“…n,k for k close to n. Thus, the behavior of TC (d) n,k for small and large k is clear. For the remaining range, there is an interesting recent conjecture for the bicombining case in [11], which has been proved for the special case of onecomponent tree-child networks in [7]. Whether this conjecture can be extended to d-combining tree-child networks is not clear yet; see the comments in Section 4.…”

“…Proof of Corollaries 1.5 and 1.7. Since the results for d = 2 are already contained in [8] (see also [7]), we can focus on the cases d ≥ 3.…”

“…where h and f are some "regular" functions. Next, we substitute s(n) = h(n)/h(n − 1) and m = κn 1/3 − 1 into (7). Then, for n → ∞ we get the expansion…”

Limit Theorems for Patterns in Ranked Tree-Child Networks

“…On the other hand, for phylogenetic networks, which are used to model reticulate evolution, very little is known about the number of occurrences of patterns when the networks from a given class are randomly sampled. This is due to the fact that even counting questions for phylogenetic networks from a given class were still open until recently; see [3,5,16,10,11,12,13,18] for progress on counting questions for some of the major classes of phylogenetic networks.…”