  We establish a log-supermodularity property for probability distributions on
binary patterns observed at the tips of a tree that are generated under any
2--state Markov process. We illustrate the applicability of this result in
phylogenetics by deriving an inequality relevant to estimating expected future
phylogenetic diversity under a model of species extinction. In a further
application of the log-supermodularity property, we derive a purely
combinatorial inequality for the parsimony score of a binary character. The
proofs of our results exploit two classical theorems in the combinatorics of
finite sets.
