Reconstructing Unrooted Phylogenetic Trees from Symbolic Ternary Metrics

Böcker and Dress (Adv Math 138:105-125, 1998) presented a 1-to-1 correspondence between symbolically dated rooted trees and symbolic ultrametrics. We consider the corresponding problem for unrooted trees. More precisely, given a tree T with leaf set X and a proper vertex coloring of its interior vertices, we can map every triple of three different leaves to the color of its median vertex. We characterize all ternary maps that can be obtained in this way in terms of 4- and 5-point conditions, and we show that the corresponding tree and its coloring can be reconstructed from a ternary map that satisfies those conditions. Further, we give an additional condition that characterizes whether the tree is binary, and we describe an algorithm that reconstructs general trees in a bottom-up fashion.