that the most probable $k$-clade (a clade of $k\ge 2$ elements), is not necessarily a clade on the species tree, even if the species tree is known to have a $k$-clade.

Undeterred by these negative results, we show in Sections \[sec:inv\] and \[sec:id\] that under the multispecies coalescent with one lineage sampled per species, the set of clade probabilities does identify the species tree topology for generic branch lengths for any number of species. The proof is based on discovering a linear combination of clade probabilities (a linear invariant) that is equal to zero for any branch lengths on any species tree with a given clade. In theory, if clade probabilities are known, it is therefore possible to identify the species tree by determining all of its clades. While this suggests a statistically consistent method of inferring a species tree from estimated clade probabilities, it remains a challenge to incorporate this insight into a practical method that outperforms greedy consensus on most finite data sets.

Finally, in Section \[sec:id\] we extend our results, in part, to cases where the species tree is non-binary and where an arbitrary number of lineages is sampled per species.

Although we frame our questions within 