# Split Probabilities and Species Tree Inference Under the Multispecies Coalescent Model

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## Abstract

Using topological summaries of gene trees as a basis for species tree inference is a promising approach to obtain acceptable speed on genomic-scale datasets, and to avoid some undesirable modeling assumptions. Here we study the probabilities of splits on gene trees under the multispecies coalescent model, and how their features might inform species tree inference. After investigating the behavior of split consensus methods, we investigate split invariants—that is, polynomial relationships between split probabilities. These invariants are then used to show that, even though a split is an unrooted notion, split probabilities retain enough information to identify the rooted species tree topology for trees of 5 or more taxa, with one possible 6-taxon exception.

### Keywords

Multispecies coalescent model Split probability Species tree identifiability### Mathematics Subject Classification

92D15## Notes

### Acknowledgements

This work was begun while ESA and JAR were Short-term Visitors and JHD was a Sabbatical Fellow at the National Institute for Mathematical and Biological Synthesis, an institute sponsored by the National Science Foundation, the US Department of Homeland Security, and the US Department of Agriculture through NSF Award #EF-0832858, with additional support from the University of Tennessee, Knoxville. It was further supported by the National Institutes of Health Grant R01 GM117590, awarded under the Joint DMS/NIGMS Initiative to Support Research at the Interface of the Biological and Mathematical Sciences.

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