ancestor of a and b , up to a consistent permutation of their elements and rows. We can then calculate P as = (diag π $^{a}$) − $^{1}$( P $^{sa}$) $^{⊺}$diag π $^{s}$, and so f ( { s, a } ). Note that the value of f ( { s, a } ) is independent of the chosen permutation of P sa by (3.1). Now fix a and calculate c = arg max$_{b}$$_{∈}$$_{T}$ f $^{b}$( { s, a } ) where f $^{b}$( { s, a } ) is the value of f for a and the most recent common ancestor of a and b , and T are the terminal nodes. The root is then the most recent common ancestor of a and c . By assumptions A, B, and C, Proposition 3.1 applies and the unrooted topology is identifiable. As for Proposition 3.1, we can also determine the value of f for every edge in the unrooted topology. As the most recent common ancestor of a and c must lie on the path from a to c , f $^{c}$( { s, a } ) then gives us the address of the root on that path. That is, we have determined the rooted topology and the value of f for every edge in the topology. We now approach the main result in Theorem 3.1. We show that a non-stationary, strand-symmetric model is fully identifiable given pairwise joint state distributions between all taxa. Up to this point we have made limited assertions about the labelling of states at internal nodes, but we will now make stronger assumptions to remove this ambiguity. One value of knowing this labelling is that it will allow us to fit continuous-time models. 3.3 Pairwise Distributions Determine the Model Start by defining two sets of matrices, the first of which is previously defined in Chang (1996). Definition 3.3. A set of matrices M is reconstructible from rows if for each M ∈ M and each permutation matrix R = I , we have RM / ∈ M . As we mentioned in the introduction, the assumption in Chang (1996) that every transition probability matrix belong to a class of matrices that is reconstructible from rows is used to identify the labelling of states at internal nodes, which otherwise 14