Author: T.DineshRaaja

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