B-trees and B+-trees are balanced tree data structures used for database indexing. B-trees have nodes with many keys and child pointers, keeping the tree height relatively small. B+-trees store all data in leaf nodes and duplicate some keys in internal nodes, allowing for sequential access. Both use splitting and combining of nodes to balance the tree during insertions and deletions.

1. Balanced Trees (B and B+)

6. Multi-way Search Tree of order of 4 Keys Pointers

7. B-Trees

9. B-Tree Properties 4. Each leaf node (other than the root node if it is a leaf) must contain at least  (m / 2)  - 1 keys. Note:  x  is the ceiling function whose value is the smallest integer that is greater than or equal to x. E.g.,  3  = 3  3.34  = 4  1.98  = 2  5.001  = 6

10. B-Tree Properties A B-tree is a fairly well-balanced tree since all leaf nodes must be at the bottom. Recall condition 2. All internal nodes (except perhaps the root node) have at least  (m / 2)  (nonempty) children  keep it bushy and balanced. Causes the tree to fan out , i.e., shorter height

12. Insertion Example Insert the following letters into what is originally an empty B-tree of order 5: C N G A H E K Q M F W L T Z D P R X Y S Order 5  max of 5 children and 4 keys. All nodes (except root) must have a minimum of 2 keys. Inserting in alphabetical order the first 4 letters:

13. Insertion Example Insert H next. No room. Split into 2 nodes. Move median item G up into new root node Insert EKQ next

14. Insertion Example C N G A H E K Q M F W L T Z D P R X Y S Inserting E, K, & Q doesn’t require splits. But inserting M does ?? - split into 2

15. Insertion Example C N G A H E K Q M F W L T Z D P R X Y S F, W, L, and T are then added without needing any split.

16. Insertion Example C N G A H E K Q M F W L T Z D P R X Y S F, W, L, and T are then added without needing any split. Adding Z requires node to split Move median (T) up & split node

17. Insertion Example Z

18. C N G A H E K Q M F W L T Z D P R X Y S D (which is the median too) Insert PRXY without any splitting

24. B Tree Deletion

25. B Tree Deletion - Delete E next - Very problematic  siblings as well as E has no extra keys - Combine the leaf with one of two siblings - Move down parent’s key that was between these two siblings

26. B Tree (Delete E)

28. B Tree Deletion

29. B+ Trees

36. Insertion Example Adding record with Key 28:

38. Insert (Leaf full; index not) 70

39. Insert (Leaf and index pages are full) Add 95  belongs here Split leaf page into 2: 75 80 and 85 90 95 Middle key (85) goes up  parent is full  split parent 25 50 60 75 85 Middle key (60) made a new parent of the parents

40. Insert (Leaf and index pages are full) 95

41. Rotation When a leaf node is full and its sibling is not. Reduce number of page splits. E.g., add 70  previously, we split the 50 55 60 65 node and brought 60 up Instead, move a record to its sibling:

43. Deletion Example Delete 70: OK since fill factor = 50% (min# records in a node)

44. Deletion Example Now delete 25: Leaf: OK (fill factor satisfied) Index: not OK  replace with 28

45. Deletion Example Delete 60: fill factor < 50%  combine leaf pages and index pages

46. Deletion Example