1. Balanced Trees

2. Binary Search tree with a balance condition Why? For every node in the tree, the height of its left and right subtrees must differ by at most 1 Ensures O(log N) depth AVL Trees

3. Must maintain balance condition Insert as in BST Follow path back to root rotating at nodes that do not meet balance condition AVL Tree Insertion

4. Cases of Imbalance Left subtree of left child Right subtree of right child External

5. Cases of Imbalance Right subtree of left child Left subtree of right child Internal

6. Single Rotation k2 k1 X Y Z k2 X ZY

7. Double Rotation k3 k1 Y X Z k3 k1 X Z k2 A B k1 XZ k3 BA Rotate k1 and k2 Rotate k2 and k3

8. Splay Trees Goal: M operations take O(MlogN) time –Some operations may take O(N) time but cost is amortized After each access, rotate node to root –Helps to balance an unbalanced tree –Next access will be fast (assumption – items are accessed frequently)

9. Zig-zag – Access X G P X Z X A B X P XZ G BA

10. Zig-zig – Access X G P Y Z X B A G P Y Z X B A

11. B-Trees Entire tree may not fit in memory 10,000,000 records, height of 25, search of 4 seconds (page 165) Want a shorter, bushier tree –M-ary tree, tree with M-way branching height of log M N

12. B-Tree Properties The data items are stored at leaves. The nonleaf nodes store up to M-1 keys to guide the searching; key i represents the smallest key in subtree i+1 The root is either a leaf or has between two and M children. All nonleaf nodes (except the root) have between ceil(M/2) and M children. All leaves are at the same depth and have between ceil(L/2) and L children, for some L.

13. Example Tree, L=5 M=5 |41| |66| |87| | | |48| |51| |54| | ||8| |18| |26| |35 ||72| |78| |83| | ||92| |97| | | | | 246246 8 10 12 14 16 18 20 22 24 26 28 30 31 32 35 36 37 38 39 41 42 44 46 48 49 50 51 52 53 54 56 58 59 66 68 69 70 72 73 74 76 78 79 81 83 84 85 87 89 90 92 93 95 97 98 99

14. Insertion If leaf is not full, add new item If leaf is full, split leaf into two each with ceil(L/2) items –also need to update parent(s) May need to split all the way up the tree –Cannot split root, but can create new root with 2 children Also, may put child up for adoption – move to neighbor if room

15. Deletion Remove item –if number of children below min, merge –percolate up tree if necessary –if root has only 1 child, remove root and make its child the new root