I was trying to solve the following problem, but don't quite know how to get started with working with the transportation metric.
Let $T = ([n], E)$ be an unweighted, undirected tree with root $r \in [n]$, and shortest path metric $d$. Let $p, q \in \mathbf{R}^{n}$ be two probability distributions over the vertices $[n]$ (i.e. $p_i \geq 0$, $\sum_i p_i = 1$). Define the transportation distance between $p$ and $q$ as $$ \tau(p, q)= \inf \left\{\sum_{i,j=1}^{n} d(i, j) w_{ij} : w_{ij} \geq 0, \sum_{i=1}^n w_{ij} = q_j, \sum_{j=1}^n w_{ij} = p_i \right\}. $$ Let $\Delta_n$ denote the space of probability distributions on $[n]$. Show that there is some integer $k$ such that the metric space $(\Delta_n, \tau)$ isometrically embeds into $(\mathbf{R}^k, d_1)$, where $$ d_1(x, y) := \sum_1^k |x_i - y_i| $$ for $x, y \in \mathbf{R}^k$.
I also don't quite know why the tree part is so relevant.