1. Fall 2007CS 2251 Self-Balancing Search Trees Chapter 9

2. Fall 2007CS 2252 Why Balance is Important Searches into an unbalanced search tree could be O(n) at worst case

3. Fall 2007CS 2253 Rotation To achieve self-adjusting capability, we need an operation on a binary tree that will change the relative heights of left and right subtrees but preserve the binary search tree property Algorithm for right rotation –Remember value of root.left (temp = root.left) –Set root.left to value of temp.right –Set temp.right to root –Set root to temp What is the algorithm for left rotation?

4. Fall 2007CS 2254 Rotation Illustrated

5. Fall 2007CS 2255 Internal Representation

6. Fall 2007CS 2256 Representation after Rotation

7. Fall 2007CS 2257 Unbalanced Trees The height of a tree is the number of nodes in the longest path from the root to a leaf node Measure the imbalance of a tree as the difference in height between the two subtrees Actual heights of the left and right subtrees are unimportant –only the relative difference matters when balancing

8. Fall 2007CS 2258 4 Types of Unbalanced Trees Left-Left –Left child is higher than right child –Left child of left child is higher than right child of left child Left-Right –Left child is higher than right child –Right child of left child is higher than left child of left child Right-Right –Right child is higher than left child –Right child of right child is higher than left child of right child Right-Left –Right child is higher than left child –Left child of right child is higher than right child of right child

9. Fall 2007CS 2259 AVL Tree Each node contains a number that represents the difference in height between the two subtrees (h R - h L ) As items are added to or removed from the tree, the balance or each subtree from the insertion or removal point up to the root is updated Rotation is used to bring a tree back into balance when the magnitude of the difference is greater than 1

10. Fall 2007CS 22510 Balancing a Left-Left Tree A left-left tree is a tree in which the root and the left subtree of the root are both left-heavy Right rotations are required

11. Fall 2007CS 22511 Balancing a Left-Right Tree Root is left-heavy but the left subtree of the root is right-heavy A simple right rotation cannot fix this Need both left and right rotations

12. Fall 2007CS 22512 To Balance Unbalanced Trees Left-Left (parent balance is -2, left child balance is -1) –Rotate right around parent Left-Right (parent balance -2, left child balance +1) –Rotate left around child –Rotate right around parent Right-Right (parent balance +2, right child balance +1) –Rotate left around parent Right-Left (parent balance +2, right child balance -1) –Rotate right around child –Rotate left around parent

13. Fall 2007CS 22513 Red-Black Trees Rudolf Bayer developed the red-black tree as a special case of his B-tree A node is either red or black The root is always black A red node always has black children The number of black nodes in any path from the root to a leaf is the same

14. Fall 2007CS 22514 Insertion into a Red-Black Tree Follows same recursive search process used for all binary search trees to reach the insertion point When a leaf is found, the new item is inserted and initially given the color red It the parent is black we are done otherwise there is some rearranging to do

15. Fall 2007CS 22515 Insertion into a Red-Black Tree Find location for new node as in the BST Color a new node red to start with Several possible cases –red parent has red sibling - recolor –red parent with no red sibling - do one or two rotations depending on location of child

16. Fall 2007CS 22516 Insertion into a Red-Black Tree

17. Fall 2007CS 22517 Insertion

18. Fall 2007CS 22518 Algorithm for Red-Black Tree Insertion

19. Fall 2007CS 22519 Non-Binary Trees Nodes can have more than two children Nodes with multiple children store multiple values

20. Fall 2007CS 22520 2-3 Trees 2-3 tree named for the number of possible children from each node Made up of nodes designated as either 2-nodes or 3- nodes A 2-node is the same as a binary search tree node A 3-node contains two data fields, ordered so that first is less than the second, and references to three children One child contains values less than the first data field One child contains values between the two data fields Once child contains values greater than the second data field 2-3 tree has property that all of the leaves are at the lowest level

21. Fall 2007CS 22521 Searching a 2-3 Tree

22. Fall 2007CS 22522 Searching a 2-3 Tree

23. Fall 2007CS 22523 Algorithm for Insertion into a 2-3 Tree

24. Fall 2007CS 22524 Inserting into a 2-3 Tree Trees are built bottom-up instead of top- down

25. Fall 2007CS 22525 Inserting into a 2-3 Tree

26. Fall 2007CS 22526 Removal from a 2-3 Tree Removing an item from a 2-3 tree is the reverse of the insertion process If the item to be removed is in a leaf, simply delete it If not in a leaf, remove it by swapping it with its inorder predecessor in a leaf node and deleting it from the leaf node

27. Fall 2007CS 22527 Removal from a 2-3 Tree

28. Fall 2007CS 22528 Removal from a 2-3 Tree (continued)

29. Fall 2007CS 22529 2-3-4 and B-Trees 2-3 tree was the inspiration for the more general B-tree which allows up to n children per node B-tree designed for building indexes to very large databases stored on a hard disk 2-3-4 tree is a specialization of the B-tree because it is basically a B-tree with n equal to 4 A Red-Black tree can be considered a 2-3-4 tree in a binary-tree format

30. Fall 2007CS 22530 2-3-4 Trees Expand on the idea of 2-3 trees by adding the 4-node Addition of this third item simplifies the insertion logic

31. Fall 2007CS 22531 Algorithm for Insertion into a 2-3-4 Tree

32. Fall 2007CS 22532 B-Trees A B-tree extends the idea behind the 2- 3 and 2-3-4 trees by allowing a maximum of CAP data items in each node The order of a B-tree is defined as the maximum number of children for a node B-trees were developed to store indexes to databases on disk storage

33. Fall 2007CS 22533 B-Trees