We describe a simple transformation that allows for the fast recovery of a tree from the probabilities such a tree induces on the colourations of its leaves under a simple Markov process (with unknown parameters). This generalizes earlier results by not requiring the transition matrices associated with the edges of the tree to be of a particular form, or to be related by some fixed rate matrix, and by not insisting on a particular distribution of colours at the root of the tree. Applications to taxonomy are outlined briefly in three corollaries.