Invariants of phylogenies in a simple case with discrete states

Under a simple model of transition between two states, we can work out the probabilities of different data outcomes in four species with any given phylogeny. For a given tree topology, if all characters are evolving under the same probabilistic model, there are two quadratic forms in the frequencies of outcomes that must be zero. It may be possible to test the null hypothesis that the tree is of a particular topology by testing whether these quadratic forms are zero. One of the tests is a test for independence in a simple 2×2 contingency table. If there are differences of evolutionary rate among characters, these quadratic forms will no longer necessarily be zero.