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? 



Recoving an unknown tree graph with knowledge of root node to leaf node distances
I think the question is a bit vague. Anyhow. You cannot recover the isomorphism type of the tree and you cannot recover the number of nodes. There is a unique tree having that set of distances with the minimal number of nodes. Namely, list the distances in increasing order
$d\_1,\dots,d\_M$. Let $r$ be an end point of a path of length $d\_1-1$.
Let $v$ be the other end point.
$r$ will be the root of the tree. $v$ gets one child, which becomes the leave of distance $d\_1$ from the root. Also, use $v$ as the root the unique minimal tree for the distances $d\_2-d\_1+1,\dots,d\_M-d\_1+1$.


This recursive definition gives you the minimal tree with this set of distances. This tree is a quotient of every tree with this set of distances. 


