This document provides an example of insertions and deletions in a B+-tree of order 1. It shows the tree being modified through 11 steps: inserting values 19 and 27, requiring splits at multiple levels; deleting values 5, 9, 12, 4, and 11 through leaf node merges and key removals up the tree; ultimately merging all nodes so the root is removed.

1. COS 597A:
Principles of
Database and Information Systems
B+-tree insert and delete
Example
1
Starting configuration
B+ tree of order d=1
root
13 30
5 10 40 50
11,12 20 30,38 60, 70
1,4 5,9 41,45
13, 18 20,29
2
1

2. Insert 19:
split leaf; expand parent with key 18
root
13 30
5 10 40 50
11,12 18 20 30,38 60, 70
1,4 5,9 41,45
13 20,29
18, 19
3
Insert 27
split leaf; expand parent with key 27 => too full
root
13 30
5 10 40 50
11,12 18 20 27 60, 70
1,4 5,9 41,45
30,38
13 20,
27,29
18, 19
4
2

3. Insert 27
split leaf; split parent;
expand grandparent with key 20 => too full
root
13 20 30
5 10 40 50
11,12 18 27 60, 70
1,4 5,9 41,45
30,38
13 20,
27,29
18, 19
5
Insert 27
split leaf; split parent; split grandparent
new root with key 20
root 20
13 30
5 10 40 50
11,12 18 27 60, 70
1,4 5,9 41,45
30,38
13 20,
27,29
18, 19
6
3

4. Delete 5, then 9
redistribute from right sibling
root
20
13 30
5 12 40 50
12 18 27 60, 70
1,4 11 41,45
30,38
13 20,
27,29
18, 19
7
Delete 12
merge leaves, delete key from parent
root
20
13 30
5 40 50
18 27 60, 70
1,4 11 41,45
30,38
13 20,
27,29
18, 19
8
4

5. Delete 4, then 11
merge leaves, delete key from parent
=>parent not full enough
root 20
13 30
40 50
18 27 60, 70
1 41,45
30,38
13 20,
27,29
18, 19
9
Delete 4, then 11
merge leaves, merge parent, bringing down key 13
=>grandparent not full enough
root 20
30
13 18 40 50
27 60, 70
41,45
1 30,38
13 20,
27,29
18, 19
10
5

6. Delete 4, then 11
merge leaves; merge parent, bringing down key 13
merge grandparent, bring down key 20,
remove root
root
20 30
13 18 40 50
27 60, 70
41,45
1 30,38
13 20,
27,29
18, 19
11
6