Set < BinarySearchTree >

• Simple translation
• Search, Insert, Remove runtimes all O(depth)

Set::Iterator < BinarySearchTree::Iterator >

• Iterator Interface: Bidirectional Iterator
• Simple Translation
• All iterator operations have constant or amortized constant runtime
• Traversal order is sorted

RedBlackTree

• Challenge goals:
1. Maintain BinarySearchTree property
2. Maintain depth = Θ(size) property
3. Cost of maintenance <= O(log size)
• RedBlackTree is a BinarySearchTree such that:
1. Every node is either Red or Black
2. The root is Black
3. Every nil-leaf is Black
4. If a node is Red then both its children are Black
5. For each node, all paths from the node to descendent nil-leaves contain the same number of Black nodes
• Challenge goals (i) and (ii) met:
1. RedBlackTree is a BinarySearchTree, by definition
2. A RedBlackTree with n non-nil nodes has height <= 2 + 2 log (n + 1)
   Proof:
1. The subtree rooted at x contains at least 2^{bh(x) - 1} - 1 internal (non-nil) nodes. (Prove by induction on height of x.)
2. Let h = height of tree. Then n = number of non-nil nodes entire tree, and:
   2^{bh(root) - 1} - 1 <= n (by (1) above)
   h <= 2bh(root) (by definition property (4))
   2^{h/2 - 1} - 1 <= n
   h/2 - 1 <= log (n + 1)
   h <= 2 + 2 log (n + 1)
   
Consequence: h = Θ(log n)

Goal (iii): Repair at cost O(log n)

• Animation