  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 more than 5 taxa, with one possible 6-taxon exception.
