Low distortion euclidean embeddings of trees

We consider the problem of embedding a certain finite metric space to the Euclidean space, trying to keep the bi-Lipschitz constant as small as possible. We introduce the notation c₂(X, d) for the least distortion with which the metric space (X, d) may be embedded in a Euclidean space It is known that if (X, d) is a metric space with n points, then c₂(X, d) ≤ 0(log n) and the bound is tight. Let T be a tree with n vertices, and d be the metric induced by it. We show that c₂(T, d) ≤ 0(log log n), that is we provide an embedding f of its vertices to the Euclidean space, such that d(x, y) ≤ ∥f(x) - f(y)∥ ≤ c log log nd(x, y) for some constant c.