  We show that every n-point tree metric admits a (1+eps)-embedding into a
C(eps) log n-dimensional L_1 space, for every eps > 0, where C(eps) =
O((1/eps)^4 log(1/eps)). This matches the natural volume lower bound up to a
factor depending only on eps. Previously, it was unknown whether even complete
binary trees on n nodes could be embedded in O(log n) dimensions with O(1)
distortion. For complete d-ary trees, our construction achieves C(eps) =
O(1/eps^2).
