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