Dimension Reduction for Finite Trees in $$\varvec{\ell _1}$$ℓ1

We show that every $$n$$n-point tree metric admits a $$(1+\varepsilon )$$(1+ε)-embedding into $$\ell _1^{C(\varepsilon ) \log n}$$ℓ1C(ε)logn, for every $$\varepsilon > 0$$ε>0, where $$C(\varepsilon ) \le O\big ((\frac{1}{\varepsilon })^4 \log \frac{1}{\varepsilon })\big )$$C(ε)≤O((1ε)4log1ε)). This matches the natural volume lower bound up to a factor depending only on $$\varepsilon $$ε. Previously, it was unknown whether even complete binary trees on $$n$$n nodes could be embedded in $$\ell _1^{O(\log n)}$$ℓ1O(logn) with $$O(1)$$O(1) distortion. For complete $$d$$d-ary trees, our construction achieves $$C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )$$C(ε)≤O(1ε2).