Complexity of the simplest phylogenetic estimation problem

The maximum-likelihood (ML) solution to a simple phylogenetic estimation problem is obtained analytically. The problem is estimation of the rooted tree for three species using binary characters with a symmetrical rate of substitution under the molecular clock. ML estimates of branch lengths and log-likelihood scores are obtained analytically for each of the three rooted binary trees. Estimation of the tree topology is equivalent to partitioning the sample space (space of possible data outcomes) into subspaces, within each of which one of the three binary trees is the ML tree. Distance-based least squares and parsimony-like methods produce essentially the same estimate of the tree topology, although differences exist among methods even under this simple model. This seems to be the simplest case, but has many of the conceptual and statistical complexities involved in phylogeny estimation. The solution to this real phylogeny estimation problem will be useful for studying the problem of significance evaluation.