CSES 1133 Tree Distances II - Core Idea

If you root the tree at node $1$, you can compute the sum of distances from node $1$ to all others using DFS. That is one answer.

Now how do you compute the answer for node $2$ (a child of $1$)? When you move the root from $1$ to $2$, some nodes get closer and some get farther.

Nodes in the subtree of $2$ get $1$ closer. Nodes outside the subtree of $2$ get $1$ farther. You can update the sum in $O(1)$ using this observation.

This is called rerooting DP.