CF917D Stranger Trees

Will shares a psychic connection with the Upside Down Monster, so everything the monster knows, Will knows. Suddenly, he started drawing, page after page, non-stop. Joyce, his mom, and Chief Hopper put the drawings together, and they realized, it's a labeled tree! A tree is a connected acyclic graph. Will's tree has $ n $ vertices. Joyce and Hopper don't know what that means, so they're investigating this tree and similar trees. For each $ k $ such that $ 0

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