Combinatorial theorems about embedding trees on the real line

We consider the combinatorial problem of embedding a tree metric into the real line with low distortion. For two special families of trees — the family of complete binary trees and the family of subdivided stars — we provide embeddings whose distortion is provably optimal, up to a constant factor. We also prove that the optimal distortion of a linear embedding of a tree can be arbitrarily low or high even when it has bounded degree.