In this paper we investigate the geometry of a discrete Bayesian network whose graph is a tree all of whose variables are binary and the only observed variables are those labeling its leaves. We provide the full geometric description of these models which is given by a set of polynomial equations together with a set of complementary implied inequalities induced by the positivity of probabilities on hidden variables. The phylogenetic invariants given by the equations can be useful in the construction of simple diagnostic tests. However, in this paper we point out the importance of also incorporating the associated inequalities into any statistical analysis. The full characterization of these inequality constraints derived in this paper helps us determine how and why routine statistical methods can break down for this model class.