# Trinets encode tree-child and level-2 phylogenetic networks

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## Abstract

Phylogenetic networks generalize evolutionary trees, and are commonly used to represent evolutionary histories of species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in trying to develop methods to construct rooted phylogenetic networks from *triplets*, that is rooted trees on three species. However, although triplets determine or *encode* rooted phylogenetic trees, they do not in general encode rooted phylogenetic networks, which is a potential issue for any such method. Motivated by this fact, Huber and Moulton recently introduced *trinets* as a natural extension of rooted triplets to networks. In particular, they showed that \(\text{ level-1 }\) phylogenetic networks *are* encoded by their trinets, and also conjectured that all “recoverable” rooted phylogenetic networks are encoded by their trinets. Here we prove that recoverable binary level-2 networks and binary tree-child networks are also encoded by their trinets. To do this we prove two decomposition theorems based on trinets which hold for *all* recoverable binary rooted phylogenetic networks. Our results provide some additional evidence in support of the conjecture that trinets encode all recoverable rooted phylogenetic networks, and could also lead to new approaches to construct phylogenetic networks from trinets.

## Keywords

Phylogenetic network Directed graph Reticulate evolution Uniqueness Encoding Trinet## Mathematics Subject Classification (2000)

68R05 05C20 92D15## Notes

### Acknowledgments

Leo van Iersel was supported by a Veni grant of The Netherlands Organisation for Scientific Research (NWO). We are grateful to the anonymous reviewers for their useful comments.

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