The Single-Root Problem (Why you need rerooting)

Suppose you want to compute, for every node $v$ in a tree, the sum of distances from $v$ to all other nodes. You already know how to do this for one fixed root using a single DFS. Root the tree at node $1$, run DFS, collect subtree sizes and distance sums.

But now you need the answer for node $2$, node $3$, and every other node. The brute force approach is to re-run DFS from each node separately. That gives $O(n)$ per node and $O(n^2)$ total. With $n = 200{,}000$, that is $4 \times 10^{10}$ operations. Far too slow.