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    $\begingroup$ You should state clearly in you hypothesis that the tree is a BST. Also, it may be easier to make the induction not on the sizes of the trees, but on the trees themselves. The inductive step would be "Suppose $T = Node(a, k, b)$ a BST where $P(a)$ and $P(b)$ holds, then…" $\endgroup$ Commented Mar 5, 2021 at 10:12