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For a vertex $u$ in a rooted tree, its subtree contains $u$ and every vertex reachable by repeatedly following children in a rooted tree, together with the connecting edges; its root of a tree is $u$. In a binary tree, the left subtree is rooted at the left child, and is empty when that child is absent; the right subtree is defined similarly. A binary search tree constrains every key in these subtrees.

The condition that each binary-tree vertex has child subtrees whose node counts differ by at most one. Recursively choosing the middle entry of an index interval establishes this condition at every vertex, with exactly one allocated node per entry. No ordering assumption on the stored values is needed for the balance property or for reproducing the input's inorder sequence.

A local change of root and child pointers that preserves inorder order. A right rotation promotes a left child, moves that child's right subtree to the old root's left position, and makes the old root its right child. The five inorder pieces remain in the same order; a left rotation reverses this transformation. Rotation does not by itself require search-tree key ordering.

A base case and a decreasing natural number measure work together. Check that every invocation satisfies its precondition, each base case finishes with the correct output, and every other invocation makes finitely many recursive calls with strictly smaller measure. All local work must also terminate. By well-founded induction, the children finish, so their caller finishes; if their answers are correct and the combination rule preserves the postcondition, the caller is correct too. This proves total correctness, not just correctness conditional on return.

Strict decrease without a lower bound is insufficient: subtracting one forever from a negative integer never reaches zero. Even a positive real-valued measure can decrease indefinitely, as $1,1/2,1/4,\dots$ shows. A well-founded relation rules out that descent. In structural recursion, a proper child subtree of a finite binary tree supplies progress without numeric arguments. The proof obligation is the relation between parent and child instances, together with terminating local work and complete base cases.