Rerooting Intuition (Reusing previous work)

Here is the core idea: when you move the root from node $u$ to its child $v$, most of the tree stays the same. Only the edge between $u$ and $v$ changes direction. Nodes in $v$'s subtree get $1$ step closer to the new root. Nodes outside $v$'s subtree get $1$ step farther.

You do not need to recompute everything from scratch. You adjust the old answer by a small amount that depends on the subtree size of $v$. This incremental update takes $O(1)$ per edge. Since a tree has $n - 1$ edges, you visit each once and get all $n$ answers in $O(n)$ total.