This document describes operations on a B-tree including insertion, deletion, splitting and merging of nodes. It shows a B-tree initially containing 26 items in a balanced structure with all leaves at the same level. Keys are added to the leaf nodes, causing splits that promote keys up the tree and rebalance the structure. Deletion is demonstrated by borrowing or merging with neighboring nodes, or demoting and promoting keys when underflow occurs.

1. B-TreesB-Trees 11

2. B-TreesB-Trees 22

3. B-TreesB-Trees 33

4. B-TreesB-Trees 44

5. B-TreesB-Trees 55
51 6242
6 12
26
55 60 7064 9045
1 2 4 7 8 13 15 18 25
27 29 46 48 53
A B-tree of order 5
containing 26 items
Note that all the leaves are at the same levelNote that all the leaves are at the same level

6. B-TreesB-Trees 66
1 2 8 12

7. B-TreesB-Trees 77
1 2
8
12 25
6, 14, 28 get added to the leaf nodes:
1 2
8
12 146 25 28

8. B-TreesB-Trees 88
Adding 17 to the right leaf node would over-fill it, so we take the
middle key, promote it (to the root) and split the leaf
8 17
12 14 25 281 2 6
7, 52, 16, 48 get added to the leaf nodes
8 17
12 14 25 281 2 6 16 48 527

9. B-TreesB-Trees 99
Adding 68 causes us to split the right most leaf, promoting 48 to the
root, and adding 3 causes us to split the left most leaf, promoting 3
to the root; 26, 29, 53, 55 then go into the leaves
3 8 17 48
52 53 55 6825 26 28 291 2 6 7 12 14 16
Adding 45 causes a split of 25 26 28 29
and promoting 28 to the root then causes the root to split

10. B-TreesB-Trees 1010
17
3 8 28 48
1 2 6 7 12 14 16 52 53 55 6825 26 29 45

11. B-TreesB-Trees 1111

12. B-TreesB-Trees 1212

13. B-TreesB-Trees 1313

14. B-TreesB-Trees 1414

15. B-TreesB-Trees 1515
1212 2929 5252
22 77 99 1515 2222 5656 6969 72723131 4343
Delete 2: Since there are enough
keys in the node, just delete it
Assuming a 5-way
B-Tree, as before...
Note when printed: this slide is animated

16. B-TreesB-Trees 1616
1212 2929 5252
77 99 1515 2222 5656 6969 72723131 4343
Delete 52
Borrow the predecessor
or (in this case) successor
5656
Note when printed: this slide is animated

17. B-TreesB-Trees 1717
1212 2929 5656
77 99 1515 2222 6969 72723131 4343
Delete 72
Too few keys!
Join back together
Note when printed: this slide is animated

18. B-TreesB-Trees 1818
1212 2929
77 99 1515 2222 696956563131 4343
Note when printed: this slide is animated

19. B-TreesB-Trees 1919
1212 2929
77 99 1515 2222 696956563131 4343
Delete 22
Demote root key and
promote leaf key
Note when printed: this slide is animated

20. B-TreesB-Trees 2020
1212
292977 99 1515
3131
696956564343
Note when printed: this slide is animated

21. B-TreesB-Trees 2121

22. B-TreesB-Trees 2222

23. B-TreesB-Trees 2323

24. B-TreesB-Trees 2424