  We consider metrical task systems on tree metrics, and present an
$O(\mathrm{depth} \times \log n)$-competitive randomized algorithm based on the
mirror descent framework introduced in our prior work on the $k$-server
problem. For the special case of hierarchically separated trees (HSTs), we use
mirror descent to refine the standard approach based on gluing unfair metrical
task systems. This yields an $O(\log n)$-competitive algorithm for HSTs, thus
removing an extraneous $\log\log n$ in the bound of Fiat and Mendel (2003).
Combined with well-known HST embedding theorems, this also gives an $O((\log
n)^2)$-competitive randomized algorithm for every $n$-point metric space.
