Retractions of Finite Distance Functions Onto Tree Metrics

Trees with positively-weighted edges induce a natural metric on any subset of vertices, however not every metric is representable in this way. A problem arising in areas of classification, particularly in evolutionary biology, is how to approximate an arbitrary distance function by such a tree metric, and thereby estimate the underlying tree that generated the data. Such transformations, from distances to tree metrics (and thereby to edge-weighted trees) should have some basic properties such as continuity, but this is lacking in several popular methods, for example (as we show) in "neighbor joining."· However, a continuous transformation, due to Buneman, frequently leads to uninteresting trees. We show how Buneman's construction can be refined so as to lead to more informative trees without sacrificing continuity.