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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 relationship with the cardinality of a certain class of words. By solving the counting problem for the words, and on the basis of the conjecture, several simple recurrence formulas for general cases arise. Moreover, a precise asymptotic analysis is provided. Our results coincide with all current formulas in the literature for particular subclasses of  tree-child networks , as well as with numerical results obtained for small networks. We expect that the study of the relationship between the newly defined words and the networks will lead to further combinatoric characterizations of this class of phylogenetic networks.

Combinatorial characterization of a certain class of words and a conjectured connection with general subclasses of phylogenetic tree-child networks