My first instinct was union-find and I went with it, but I almost forgot to validate that parent values themselves are in range before doing anything else.
Clarify the definition of a valid rooted tree: exactly one root, every node except the root has exactly one parent, and no cycles. Then propose an algorithm that counts roots, validates parent references, and detects cycles, discussing time and space complexity.
Pro tip: Mention that you can solve it in O(n) time and O(n) space using an array to track visited nodes, and that you can optimize space by marking visited nodes in-place if mutation is allowed. Also, explicitly handle edge cases like empty array, multiple roots, and self-loops.
Confirm the definition of a valid rooted tree and ask about constraints (e.g., array size, possible values). Discuss edge cases: empty array, multiple roots, cycles, disconnected components, and invalid parent indices.
Propose an algorithm that checks: (1) exactly one root (-1), (2) every other node has a valid parent index, (3) no cycles, and (4) all nodes are connected to the root. Use DFS/BFS or union-find for cycle detection.
State that the algorithm runs in O(n) time and O(n) space. Discuss alternative approaches (e.g., union-find, iterative marking) and their trade-offs.
Trace the algorithm on a small example, such as [-1, 0, 0, 1] (valid) and [-1, 0, 1, 0] (cycle), to demonstrate correctness.
Reiterate the key conditions for a valid tree and confirm that the algorithm handles all cases efficiently.
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