Method 2 – Single DFS with Entry/Exit Times

Intuition

By using a single DFS traversal, we can record entry and exit times for each node, which allows us to quickly determine ancestor-descendant relationships. We also compute the XOR of each subtree during this traversal. This structure enables us to efficiently enumerate all valid ways to split the tree by removing two edges and calculate the resulting component XORs.

Approach

Perform a DFS from the root, recording for each node its entry (in[x]) and exit (out[x]) times, and the XOR of its subtree (sum[x]).
For every pair of non-root nodes u and v, consider removing the edge to their parent for each.
Use the entry/exit times to check if one node is an ancestor of the other, and compute the three component XORs accordingly:
- If u is ancestor of v: parts are sum[0]⊕sum[u], sum[u]⊕sum[v], sum[v].
- If v is ancestor of u: parts are

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