I was looking at this operation in a tree, and try to relate it to the diameter of the tree.
Pick a path of length $m$, so let it be $v_1v_2\ldots v_mv_{m+1}$. Remove all the edges in the path, and add edge $v_1v_j$, where $2 \leq j\leq m+1$. It is easy to see the resulting graph is still a tree.
Intuitively, it replace a long path with many smaller paths.
I conjecture the longest path in a tree of $n$ vertices after doing this operation $k$ times, $i$th time act on a path of length $l_i$, is at most $n-\sum_{i=1}^k l_i + k$.
I have a hard time proving it or show a counterexample, because it is not easy to come up with a characterization of the tree after each operation. It is possile the longest path does not change at all in a sequece of operations.
I hope to find pointers to anything known about this kind of operations.