So, finding a key takes O(height) time, how much time does it take to find all nodes with a key greater than a given key? What are the constant factors?

If it will be done properly, you would probably find the key, and then go In-Order to the next ones.

So it will be O(logn) + m . Where m is the number of bugs greater than the key.

Worst case would be O(logn) + n = O(n)

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