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The evolution of aligned DNA sequence sites is generally modeled by a Markovprocess operating along the edges of a phylogenetic tree. It is well known thatthe probability distribution on the site patterns at the tips of the treedetermines the tree and its branch lengths. However, the number of patterns istypically much larger than the number of edges, suggesting considerableredundancy in the branch length estimation. In this paper we ask whether theprobabilities of just the `edge-specific' patterns (the ones that correspond toa change of state on a single edge) suffice to recover the branch lengths ofthe tree, under a symmetric 2-state Markov process. We first show that thisholds provided the branch lengths are sufficiently short, by applying theinverse function theorem. We then consider whether this restriction to shortbranch lengths is necessary, and show that for trees with up to four leaves itcan be lifted. This leaves open the interesting question of whether this holdsin general.

Tree split probabilities determine the branch lengths