Is there any result on approximating an arbitrary tree metric by an HST metric?
The distortion of embedding any $n$-point metric in  ultrametric is at most $n-1$, and on the other hand, the distortion of embedding the path metric $P_n$ in ultrametric is at least $n-1$.

Similarly, if you are interested in probabilistic embedding, then by Fakcharoenphol, Rao, and Talwar result mentioned above, any $n$-point metric space probabilistically embeds in ultrametric with distortion $O(\log n)$. 
On the other hand, the distortion of probabilistically embedding $P_n$ in ultrametric is $\Omega(\log n)$.