because where π r and π s are the marginal probability vectors at r and s respectively, so that the sign of det P rs becomes irrelevant. The result in Buneman (1971) shows that if the values of an additive function are known between the pairs of all terminal nodes, then under the conditions of Proposition 3.1, the topology and the value of the function on all edges of the topology are determined. As f can be calculated from pairwise distributions between terminal nodes, so the topology and the value of f on every edge can be determined. Corollary 3.1. Suppose the evolutionary tree has nodes of degree three or one, with the exception of one special node which we designate the root. The root can have degree one, two, or three. Assume that Assume also that for each edge { u, v } , where v is further from the root than u , Then the rooted topology of the tree is recoverable from pairwise distributions of states at terminal nodes. That is, if two models in the family induce the same pairwise distributions of character states at the terminal nodes, then the topologies recovered by those two models must be equivalent. Proof. Under assumptions A, B, and D, Lemma 3.1 can be applied to any pair of terminal nodes a and b to obtain π s and P $^{sa}$, where s is the most recent common 13 (B) P uv is invertible, (A) π m is compositionally asymmetric for every internal node m . (C) P uv is not a permutation matrix, and (D) P uv is strand-symmetric. det ( P $^{rs}$P $^{sr}$) = det ( (diag π $^{r}$) − $^{1}$( P $^{sr}$) $^{⊺}$diag π $^{s}$P $^{sr}$) ≥ 0 ,