1. Data Structures and Algorithms1 B-Trees with Minimum=1 2-3 Trees

2. Data Structures and Algorithms2 Delete (T, X, success) /*Delete from tree T the item with key X. The operation fails if no such item. The flag success indicates whether the operation succeeds*/ //Attempt to locate I with search key X IF I is present THEN swap item I into leaf L which contains the inorder successor of I /* deletion begins from leaf L */ IF L has no items THEN Fix (L) success := true ELSE success := false

3. 3 Fix (N) /* N is a node with no item. Note, if N is an internal node, then it has one child. */ Let P be the parent of N. If N is the root, delete it and return. IF some sibling of N has two items THEN distribute items among N, the sibling, and P IF N is internal THEN move the appropriate child from the sibling to N ELSE /* must merge the node */ Choose an adjacent sibling S of N Bring the appropriate item down from P into S If N is internal THEN Move N’s child to S Delete node N If P is now without an item THEN Fix (P)

4. Data Structures and Algorithms4 Insert (T, newitem) /* Insert newitem into tree T */ Let X be the search key of new item Locate the leaf L in which X belongs Add newitem to L IF L now has three items THEN Split (L)

5. 5 Split (N) /* Split node N which contains 3 items. Note that if N is internal then it has 4 children */ Let P be the parent of N /* if N is the root, then create a new node P */ Replace node N by two nodes, N1 and N2 Give N1 the item in N with the smallest search key value Give N2 the item in N with the largest search key value If N is an internal node THEN N1 becomes the parent of N’s two leftmost children N2 becomes the parent of N’s two rightmost children Send up to P the item in N with the middle search key value If P now has 3 items THEN Split (P)

6. Data Structures and Algorithms6 Insertion Given50 30 70,90 10,20 40 60 80 100 Insert 39 Insertions are always at a leaf 50 30 70,90 10,20 39,40 60 80 100

7. Data Structures and Algorithms7 50 30 70,90 10,20 38,39,40 60 80 100 Insert 38 illegal 50 30,39 70,90 10,20 38 40 60 80 100

8. Data Structures and Algorithms8 Insert 37 50 30,39 70,90 10,20 37,38 40 60 80 100 When the height grows it does so from the top.

9. Data Structures and Algorithms9 Insert 36 50 30,39 70,90 10,20 36,37,38 40 60 80 100 illegal 50 30,37,39 10,20 36 38 40 illegal 37,50 30 39 10,20 36 38 40

10. Data Structures and Algorithms10 Insert 35, 34, 33 37,50 30 10,20 35,36 37,50 30 10,20 34,35,36 h illegal 37,50 30,35 10,20 34 36 ALL leaves are at the same level

11. Data Structures and Algorithms11 h 37,50 30,35 39 70,90 10,20 33,34 36 38 40 60 80 100

12. Data Structures and Algorithms12 Given 50 30 70,90 10,20 40 60 80 100 Delete 50 60 30 70,90 10,20 40 80 100 60 30 90 10,20 40 70,80 100 Deletion

13. Data Structures and Algorithms13 Delete 100 60 30 90 10,20 40 70,80 60 30 80 10,20 40 70 90

14. Data Structures and Algorithms14 Delete 60 70 30 80 10,20 40 90 70 30 10,20 40 80,90 N 30 70 10,20 40 80,90 30,70 10,20 40 80,90

15. Data Structures and Algorithms15 Delete 70 30,80 10,20 40 90 Delete 80 30,90 10,20 40 30 10,20 40,90

16. Data Structures and Algorithms16 Given 50 30 70,90 10,20 40 60 80 100 Delete 70 You always begin deletion from a leaf so swap with inorder successor. 50 30 80,90 10,20 40 60 70 100 illegal 50 80,90 60 100

17. Data Structures and Algorithms17 50 90 60,80 100 50 30 90 10,20 40 60,80 100

18. Data Structures and Algorithms18 Delete 100 50 90 60,80 This leaf can spare a value 50 80 60 90 50 30 80 10,20 40 60 90

19. Data Structures and Algorithms19 Delete 80 50 30 90 10,20 40 60 80 50 30 90 10,20 40 60 50 30 10,20 40 60,90 Can‘t spare a value 30,50 10,20 40 60,90