Lets $T=(V,E,W)$ be a weighted tree (undirected acyclic graph) with positive weights on $n$ nodes. The weights define a natural metric on the set $V$ : $d(i,j) = $ weight of the (unique) path between $i$ and $j$ in $T$.

Now, lets suppose that $T$ is unknown and that we have access only to the $n\times n$ distance matrix induced by the tree. My question is : Can one learn the structure of $T$ without looking at all the ${n\choose 2}$ distances?

Alternatively, it is easy to see that the tree $T$ is the MST of the complete graph on $V$ with weights given by $d(\cdot,\cdot)$. Is there a sub-linear algorithm for finding the MST of this special, completely connected graph?

Edit: I must add that I would be happy to restrict attention to certain families of trees. For instance, this can be done when the tree $T$ is a line graph and if we know this before hand.