For a binary tree, you'd prove theorems inductively by proving them on leaf nodes, then proving that if both of an interior node's children satisfy the theorem, the interior node must as well.

Do you see any value though in the 'random' structure part? Or do you think that to prove it on a leaf, then on a interior node would suffice to say that it works? I can see the theoretical beauty of that, but real-world exp. tells me that edge cases are not that easily revealed and running it through several random iterations would be a good idea.

Nice job on the root node, btw.

Thanks.