## Inductive proof that subtree at node x contains at least 2bh(x) - 1 internal nodes

- Inductive step: x has positive height and 2 children
- Each child has black-height of bh(x) or bh(x)-1 (Why?)
- The height of a child = (height of x) - 1
- So the subtrees rooted at each child contain at least 2bh(x) - 1 - 1 internal nodes
- Thus subtree at x contains (2bh(x) - 1 - 1) + (2bh(x) - 1 - 1) + 1= 2•2bh(x)-1 - 1 = 2bh(x) - 1 nodes

- Inductive step: x has positive height and 2 children

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