we also relax an important assumption that Chang (1996) makes about the structure of the model. As noted in Zou et al. (2011) and addressed in Mossel and Roch (2006) the model of Barry and Hartigan (1987a) is only identifiable up to an arbitrary relabelling of states at internal nodes of the phylogeny. Chang (1996) addresses this problem by assuming that the transition probability matrix for every edge in the topology is reconstructible from rows . As we shall demonstrate, this assumption can be restrictive in practice, so, motivated by Remark 4 in Chang (1996), we partially relax it. In Section 2 we briefly introduce the necessary notation and theoretical context. Section 3 contains the main results of the paper, where we prove that the full topol- ogy and parameters of a discrete-time, non-stationary, strand-symmetric Markov model can be recovered from the pairwise joint probability distributions of states between extant taxa. In this section we also extend the result to continuous-time models which are used more commonly in practice. Section 4 proves that the results in Section 3 provide the necessary basis for consistent statistical estimation of the models in question for multiple sequence alignments of increasing length. Section 5 gives some concluding remarks. 2 Markov Models on Trees: Definitions and No- tation We consider a finite set of extant taxa T whose phylogenetic history we wish to infer. The history is modelled as a tree which is a set of nodes S which represent extant or ancestral species and undirected edges E which represent genetic descent. The edges are a set of unordered pairs of nodes, so if { r,s } ∈ E , an edge exists between nodes r and s . The nodes consist of the terminal nodes , which are just T , and the internal nodes N , so that N = S \ T . The internal nodes represent ancestral taxa 6