Is there any result on approximating an arbitrary tree metric by an HST metric? Is there any known result on approximating an arbitrary tree metric by an HST metric (or an Ultrametric)? What is the distortion? Thanks.
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 $\Omega(\log n)$.