  We introduce a notion of finite sampling consistency for phylogenetic trees
and show that the set of finitely sampling consistent and exchangeable
distributions on n leaf phylogenetic trees is a polytope. We use this polytope
to show that the set of all exchangeable and infinite sampling consistent
distributions on 4 leaf phylogenetic trees is exactly Aldous' beta-splitting
model and give a description of some of the vertices for the polytope of
distributions on 5 leaves. We also introduce a new semialgebraic set of
exchangeable and sampling consistent models we call the multinomial model and
use it to characterize the set of exchangeable and sampling consistent
distributions.
