Imagine I have an unknown (undirected) tree graph, $G$, with some unknown number of nodes $||V||$. However, I know the edge-length between nodes is of fixed size, $L_{edge} = 1$, and I have access to the set of distances $(d_1, ..., d_i, ..., d_M)$ between a root node, $v_{root}$, and leaf nodes, $l_i$.

When is this limited information sufficient to recover the structure of $G$, and for arbitrary $G$, beyond tree depth, what inferences might this allow?