# Tree-based networks: characterisations, metrics, and support trees

## Abstract

Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer. One way to obtain such a network is by starting with a (rooted) phylogenetic tree *T*, called a base tree, and adding arcs between arcs of *T*. The class of phylogenetic networks that can be obtained in this way is called tree-based networks and includes the prominent classes of tree-child and reticulation-visible networks. Initially defined for binary phylogenetic networks, tree-based networks naturally extend to arbitrary phylogenetic networks. In this paper, we generalise recent tree-based characterisations and associated proximity measures for binary phylogenetic networks to arbitrary phylogenetic networks. These characterisations are in terms of matchings in bipartite graphs, path partitions, and antichains. Some of the generalisations are straightforward to establish using the original approach, while others require a very different approach. Furthermore, for an arbitrary tree-based network *N*, we characterise the support trees of *N*, that is, the tree-based embeddings of *N*. We use this characterisation to give an explicit formula for the number of support trees of *N* when *N* is binary. This formula is written in terms of the components of a bipartite graph.

## Keywords

Phylogenetic network Tree-based network Nonbinary Matching Support tree Bipartite graph## Mathematics Subject Classification

05C05 05C85 92D15## Notes

### Acknowledgements

We would like to thank Momoko Hayamizu for helpful comments concerning Theorem 8 and the two anonymous reviewers for further suggestions. JCP thanks the Biomathematics Research Centre of the University of Canterbury (and especially MS and CS) for hosting his visit, which led to this collaboration. We thank the (former) Allan Wilson Centre and the New Zealand Marsden Fund for funding support for this project.

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