Dimension Reduction for Finite Trees in ℓ 1

We show that every n-point tree metric admits a (1 + ε)-embedding into ` C(ε) logn 1 , for every ε > 0, where C(ε) ≤ O ( ( 1ε ) 4 log 1ε ) ) . This matches the natural volume lower bound up to a factor depending only on ε. Previously, it was unknown whether even complete binary trees on n nodes could be embedded in ` O(logn) 1 with O(1) distortion. For complete d-ary trees, our construction achieves C(ε) ≤ O ( 1 ε2 ) .