Recent Progress in Phylogenetic Combinatorics

of D is an R-tree. (ii) There exists a tree (V,E) whose vertex set V contains X, and an edge weighting ` : E → R that assigns a positive length `(e) to each edge e in E, such that D is the restriction of X to the shortest-path metric induced on V. (iii) There exists a map w : S(X) → R≥0 from the set S(X) of all bi-partitions or splits of X into the set R≥0 of non-negative real numbers such that, given any two splits S = {A,B} and S′ = {A′, B′} in S(X) with w(S), w(S′) 6= 0, at least one of the four intersections A ∩A′, B ∩A′, A ∩B′, and B ∩B′ is empty and D(x, y) = ∑ S∈S(X:x↔y) w(S) holds where S(X : x↔y) denotes the set of splits S = {A,B} ∈ S(X) that separate x and y. (iv) D(x, y)+D(u, v) ≤ max ( D(x, u)+D(y, v), D(x, v)+D(y, u) ) holds for all x, y, u, v ∈ X