  We study the Wasserstein (or earthmover) metric on the space $P(X)$ of
probability measures on a metric space $X$. We show that, if a finite metric
space $X$ embeds stochastically with distortion $D$ in a family of finite
metric trees, then $P(X)$ embeds bi-Lipschitz into $\ell^1$ with distortion
$D$. Next, we re-visit the closed formula for the Wasserstein metric on finite
metric trees due to Evans-Matsen \cite{EvMat}. We advocate that the right
framework for this formula is real trees, and we give two proofs of extensions
of this formula: one making the link with Lipschitz-free spaces from Banach
space theory, the other one algorithmic (after reduction to finite metric
trees).
