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1.Binary Search Tree and AVL Tree Is this a binary search tree? • Is this the only BST with the months of the year? • Can you draw three binary search trees with the keys 12, 41, 9, 55, 36?

2.AVL trees (Inventors G.M. Adelson-Velsky and E.M. Landis) An AVL tree satisfies the height-balance property: For any tree rooted at an internal node n, the heights of the left and right subtrees differ by at most 1. (a)(d) satisfy height-balance property, (b) and (c) don’t

3.Examples of AVL Trees 4 44 2 3 17 78 1 2 1 0 32 50 88 0 0 0 0 1 1 48 62 0 0 0 0 h+3 h+2 h+1 h h+1 T1 h-1 h T4 T2 T3 Valid AVL trees - satisfy height-balance property

4.Height of an AVL tree Is h= O(log n) for AVL trees too? Yes! Why? Let n(h) be the smallest number of internal (i.e. non- leaf) nodes of height h in an AVL tree. Then n(1) = 1,n(2) = 2 , and n(h) = n(h − 1) + n(h − 2) + 1 So, n(h) > 2.n(h − 2) , and n(h) > 2 i.n(h − 2i) . h −1 Substitute i = . This leads to 2 h−1 h−1 n(h) > 2 2 .n(h − (h − 1)) = 2 2 h −1 Therefore, log n(h) > 2 So h < 1+ 2 log n(h) , and the number of leaves ≤ 1+number of internal nodes .

5.Insertion algorithms for AVL Trees 1. First insert the key w into the BST. If it maintains the height-balance property then fine. Otherwise we need to rebalance it. 3. Re-balance the tree by performing appropriate rotations as shown below to repair the tree. y x right rotation x y T3 T1 Left rotation T1 T2 T2 T3

6.Rotation Operations for rebalancing Starting from w, travel up and find the first unbalanced node. Let z be the first unbalanced node, y be the child of z that comes on the path from w to z, and x be the grandchild of z that comes on the path from w to z. There can be four possible cases that need to be handled as x, y and z can be arranged in 4 ways. Following are the possible 4 arrangements: Four different cases 1. y = left child of z, x = left child of y (Right rotation) 2. y = left child of z, x = right child of y (Left Right) 3. y = right child of z, x = right child of y (Left rotation) 4. y = right child of z, x = left child of y (Right Left)

7.y = left child of z, x = left child of y (Right rotation) y = left child of z, x = right child of y (Left Right)

8.y = right child of z, x = right child of y (Left rotation) y = right child of z, x = left child of y (Right Left)

9.Here is an example 44 z 44 z 17 78 78 17 y y 50 88 62 88 32 32 48 62 x x left rotate (50) 50 54 48 54 inserted 44 y right rotate (78) 17 62 z x 50 78 32 48 54 88 For the delete operation, follow the usual delete algorithms for BST and then, if needed, restore the height-balance property.

10.Red-Black trees Another approach to keeping a binary search tree balanced. Contains two types of nodes: red and black. By definition, all null entries (non-existent children) are black. Maintains three invariants: 1. The root is always black. 2. A red node always has black children. 3. The number of black nodes in any path from the root to a leaf (a. k. a. black depth) is the same. The following are NOT red-black trees. Why?

11. The following ARE valid red black trees. 10 10 5 15 5 15 1. Why do we need to learn about this? Because h ≤ 2 log (n+1): It guarantees search, insert, delete in O(log n) time. 2. But isn’t that true for AVL trees too? Yes, but in AVL trees, there are more rotations on an average (up to O(log n) in the worst case) during insert operations. In contrast Red-Black trees need at most one rotation per insertion and at most two rotations for deletion, so they work better with frequent insert and delete operations.

12.3. Search in Red-black trees For AVL trees, h ≤ 1.44 log n For RB trees, h ≤ 2 log(n) So lookup is faster in AVL trees, particularly for large n, but it comes at the expense of slower insertion and deletion times. 3. The TreeMap class TreeMap class implements the Map interface using a Red-Black tree. The binary tree sorts the keys in the ascending order. The HashMap class also implements a Map interface, but is an unordered collection of key-value pairs.

13.Inserting a node Follow the normal rules of insertion in a BST. Let x = node to be inserted, and y = parent of x. Make the new node x a red node. Two sample cases: Case 1. The sibling s of y is black Case 2. The sibling s of y is red . If 30 is the root, then color it black.