Largest subtree sum for each vertex of given N-ary Tree

Compute subtree sums with postorder DFS. For a node, recursively obtain each child's subtree sum, add them to the node's own value, then return that total to its parent. At that moment the total is exactly the sum of all vertices in the node's rooted subtree, so compare it with a global best and record the node if larger. If the input is an undirected n-ary tree, pass the parent into DFS or maintain visited state so the traversal does not walk back up an edge. Clarify tie behavior, such as keeping the first smallest-index node, because equal maximum sums are possible. Negative node values matter: initialize the best to negative infinity rather than zero. For very deep trees, an iterative postorder traversal can avoid call-stack overflow. The proof is direct structural induction on children. Each edge and node is visited once, giving O(n) time and O(n) space for adjacency plus recursion or an explicit stack.