Why non-uniform priors on clades are both unavoidable and unobjectionable

Pickett and Randle (2005) showed that for n > 4 taxa, any prior probability distribution that is uniform with respect to tree topologies induces a non-uniform distribution on clades. Steel and Pickett (2006) then strengthened this result by proving that any label invariant distribution on trees induces a non-uniform prior on clades, but left open the possibility of a uniform distribution on clades that is not label invariant. The point of this letter is to show that uniform priors on clades are impossible on any probability distribution on topologies (with five or more leaf taxa) and that this fact is in no way a bad consequence for Bayesian phylogenetic practice.