An analytical upper bound on the number of loci required for all splits of a species tree to appear in a set of gene trees

Background: Many methods for species tree inference require data from a sufficiently large sample of genomic loci in order to produce accurate estimates. However, few studies have attempted to use analytical theory to quantify "sufficiently large". Results: Using the multispecies coalescent model, we report a general analytical upper bound on the number of gene trees n required such that with probability q, each bipartition of a species tree is represented at least once in a set of n random gene trees. This bound employs a formula that is straightforward to compute, depends only on the minimum internal branch length of the species tree and the number of taxa, and applies irrespective of the species tree topology. Using simulations, we investigate numerical properties of the bound as well as its accuracy under the multispecies coalescent. Conclusions: Our results are helpful for conservatively bounding the number of gene trees required by the ASTRAL inference method, and the approach has potential to be extended to bound other properties of gene tree sets under the model.