Low-distortion embeddings of trees

We prove that every tree T = (V, E) on n vertices can be embedded in the plane with distortion O(rootn); that is, we construct a mapping f: V --> R-2 such that rho(u,v) less than or equal to parallel tof(u) - f(v)parallel to less than or equal to O(rootn) (.) p(u,v) for every u, v is an element of V, where p(u, v) denotes the length of the path from u to v in T (the edges have unit lengths). The embedding is described by a simple and easily computable formula. This is asymptotically optimal in the worst case. We also prove several related results.