Then the full model is identifiable. That is, the topology and all of the transition probability matrices are uniquely determined by the joint distribution of character states at the terminal nodes of the tree. Remark 3.1. The relaxation of the assumption that every transition probability matrix must be reconstructible from rows to the assumption that every matrix must be sympathetic to those which are might not seem important, but in fact it greatly increases the range of models to which the theorem applies. By allowing sympathetic transition probability matrices, we are in effect allowing long branches in the tree as long as there are sufficient short branches to consistently reconstruct the labelling of states at internal nodes. Proof of Theorem 3.1. We have from Corollary 3.1 the full rooted topology. It is enough to then determine a transition probability matrix for each edge. We will proceed by induction. If the topology has two terminal nodes and one node is the root then the full model is trivially determined. If the topology has two terminal nodes and neither node is the root, then by assumptions A, B, and D we can apply Lemma 3.1 to recover the model up to a permutation of labels of internal states. At least one transition probability matrix must be reconstructible from rows by assumption E and the other at worst will not give an alternative ordering by assumption F, so the ordering of states at the internal node can be recovered and the full model is determined. If the topology has more than two terminal nodes, choose an arbitrary terminal node a and denote its neighbouring node m . We treat two cases separately. If a is not the root, choose another terminal node b such that m is the most recent common ancestor of a and b . By assumptions A, B, and D, we can apply Lemma 3.1 to a and b to determine P ma up to a permutation of labels of internal states. By assumption F we can deduce whether P ma ∈ M , and if it is we can infer 17