E$_{2}$ are obtained by applying the function to the nodes in the edges E$_{1}$ . That is, the topologies T$_{1}$ and T$_{2}$ are equivalent if they are the same up to a possible relabeling of internal nodes. The following is Proposition 3.1 in Chang (1996): Proposition 3.1. Consider a family of Markov models satisfying the following con- ditions: Then the unrooted topology is identifiable from the joint distributions of character states at pairs of terminal nodes. That is, if two models in the family induce the same pairwise distributions of character states at their terminal nodes, then the topologies of those two models must be equivalent. As stated in Chang (1996), Proposition 3.1 follows by combining a result in Buneman (1971) with the additive function defined on pairs of nodes as The function is additive in the sense that f ( { r, s } ) = ∑ e ∈ path( { r,s } $_{)}$f ( e ), where path( { r, s } ) is the set of edges joining any two nodes r and s . Chang attributed the log det distance to Barry and Hartigan (1987b) and Cavender and Felsenstein (1987). It is unclear whether he was aware that the above definition of f is equivalent to the paralinear genetic distance measure introduced by Lake (1994). In any case we will prefer to formulate this function almost equivalently as 12 f ( { r, s } ) = − log det ( P $^{rs}$P $^{sr}$) , (3.1) f ( { r, s } ) = − log det P rs − log det P $^{sr}$. 1. The edge transition matrices are invertible and not equal to a permutation matrix. 2. There is a node v with π $^{v}$( i ) > 0 for each i ∈ C , that is, each character state has positive marginal probability at v .