Boht tej ho rahe ho, hein!

"Why don't you have problems on Trees in Convex Hull?" Fachin always keeps asking us to give problems on trees. We gave him the question of finding the center of a node-weighted tree, but he misunderstood the question and solved the following problem: Given a tree on $N$ nodes with weights $w_i$ on the $i^{\text{th}}$ node, find the vertex $v$ such that the sum of weighted distances from $v$ to a