the correct permutation to recover P $^{ma}$. Otherwise, restart this iteration with a different a . 

If a is the root, choose two other terminal nodes b and c such that the most recent common ancestor of b and c is m . Again by assumptions A, B, and D, we can apply Lemma 3.1 to b and c to obtain a stochastic matrix V and a diagonal matrix U such that P mb = RV and diag π m = RUR ⊺ for some permutation matrix R . Then P ma = RU − $^{1}$( V − $^{1}$) $^{⊺}$J $^{ba}$. Once more if P ma ∈ M , we infer the correct matrix R and we know P $^{ma}$. Otherwise, restart this iteration with a different a . 

So we are able to determine P ma for a terminal node a and its neighbour m . The induction step is then to remove the edge { m,a } from the tree, and make m a terminal node in two subtrees, or one subtree if m happens to be the root. For any subtree, J im = J $^{ia}$( P $^{ma}$) − 1 for any terminal node of the subtree i . This step can be repeated until all subtrees have two taxa and the full model is recovered on all edges. 

## 3.4 Continuous-Time Models 

We will now show that Theorem 3.1 is still applicable in almost all cases if we further restrict the model to be continuous-time. We state this explicitly as some care is required because the matrix logarithm can have multiple roots. Continuous- time models provide more information than discrete-time models, for instance for questions concerning genetic distance and relative rates of evolution. 

Theorem 3.2. Make the assumptions of Theorem 3.1. Additionally assume that for every { u,v } ∈ E , where v is further from the root than u , there exists a unique mapping Q uv = log P uv where every off-diagonal element of Q uv is non-negative. Also replace assumption D with the assumption that Q uv is strand-symmetric. Then the full model is identifiable from the joint distribution of states at the terminal nodes of the tree. 

