  One approach to estimating a species tree from a collection of gene trees is
to first estimate probabilities of clades from the gene trees, and then to
construct the species tree from the estimated clade probabilities. While a
greedy consensus algorithm, which consecutively accepts the most probable
clades compatible with previously accepted clades, can be used for this second
stage, this method is known to be statistically inconsistent under the
multispecies coalescent model. This raises the question of whether it is
theoretically possible to reconstruct the species tree from known probabilities
of clades on gene trees. We investigate clade probabilities arising from the
multispecies coalescent model, with an eye toward identifying features of the
species tree. Clades on gene trees with probability greater than 1/3 are shown
to reflect clades on the species tree, while those with smaller probabilities
may not. Linear invariants of clade probabilities are studied both
computationally and theoretically, with certain linear invariants giving
insight into the clade structure of the species tree. For species trees with
generic edge lengths, these invariants can be used to identify the species tree
topology. These theoretical results both confirm that clade probabilities
contain full information on the species tree topology and suggest future
directions of study for developing statistically consistent inference methods
from clade frequencies on gene trees.
