Embeddings of Lipschitz-free spaces into $\ell_1$

In this note we study the Lipschitz-free space $\mathcal{F}(M)$ for a metric space $M$ which is a subset of an $\mathbb{R}$-tree. The main result states that $\mathcal{F}(M)$ embeds almost-isometrically into $\ell_1$ whenever the metric space $M$ is complete, separable and has length measure 0. If moreover $M$ is a proper metric space, then the converse is also true. We also prove that, for any subset $M$ of an $\mathbb{R}$-tree, every extreme point of the unit ball of $\mathcal{F}(M)$ is an element of the form $(\delta(x)-\delta(y))/d(x,y)$ for $x\neq y\in M$.