B+ trees are a data structure used to store sorted data like files in a disk. Each node contains key values and pointers to other nodes. Leaf nodes contain file data while internal nodes contain keys to guide searching. Insertion may cause nodes to split, requiring redistribution of keys and merging of nodes. Deletion is handled through redistribution or merging of neighboring nodes to maintain a minimum number of keys per node. B+ trees provide efficient storage and retrieval of sorted data through balanced tree structure and localized rebalancing during updates.

1. Topic
B+ TREES
Presented by:
Mujahid Hussain Jalbanee
(Student of Software Engineering)

2. Tree:
 Tree is collection of nodes.
 Nodes have data and pointers.
 Pointers can be dynamic and can be thousands.
 Files are just like leaves.
 1st node called Root.
 Sibling nodes have same parent node.
 Path means to go towards particular node.

3. B+ Trees:
 Every node has pointers to other nodes and key values.
 If there are N pointers then there can be N-1 number of key
values.
 Key values are repeated from root node to leaf node because
data pointers are located at leaf nodes.

4. B+ Trees Structure.
A B
Values < A
Values >=A && <B
Values >= B

5. B+ Tree Insertion:
 Lets take the following elements
2, 5, 7, 10, 13, 16, 20, 22, 23, 24….
 Let be number of pointer, indirectly, 3 key values can be there in
a node at a maximum case….
 Insert 2,5,7….
 Now insert 10….
2 5 7

6. Continue……..
 When we insert 10 then node will be break down into two..
 After inserting 10..
 There should be a parent to look after when there nodes > 1, so
copy 7 into parent node….
2 5 7 10

7. Lets see….
Insert 13, 16….. Node again breaks
7
2 5 7 10
Continue……

8.  After inserting 13 and 16….
 Nodes over flows….
 After insertion….
7 10 13 16
2 5 13 16
7 13
7 10
Continue…..

9. Insert 20 and 22
Violation… above 3 keys… again break up… 13 and 16 in one 20
and 22 will be in another new node… and then should copy in the
parent node….
13 16 20 22
2 5 7 10 13 16 20 22
7 13 20

10.  Now lets insert 23 and 24… we can directly insert 23….
7 13 20
2 5 7 10 13 16 20 22 23

11.  Now insert 24…. Node over flows…
 So, we should break up and copy 23 to the parent.. But this
time parent node is over flows..
 Now break down parent node and move the smallest element
from second partition to newly created parent node…
20 22 23 24

12.  After inserting 24….
13
7 20 23
2 5 23 2420 2213 167 10

13.  When number of search-key values < (n/2)-1
LEAF NODE
 Redistribute to sibling
Right node not less than left node
Replace the between-value in parent by their smallest value of
right node…
Merge (contain too few entries)
Move all values, pointer to left node
Remove the between-value in parent…
B+ Trees Deletion

14. 13 18
9 13
13 14 169 10
14 16
14 18
 Now delete 10….
After Deletion…..

15. Now delete 10….
 After deletion….
9 10 13 14
18 22
13 18 22
9 13 14

16.  Non-Leaf Node..
Redistribute to sibling
Though parent
Right node not less than left node
Merge(contain too few entries)
Bring down parent
Move all values, pointers to left node
Delete the right node, and pointers in parent….

17. 5 20
83
31 Some sub tree
Now delete 3, where n=3….

18. After deletion 3….
20
5 8
1
Some sub tree

19. Advantages:
Automatically adjust the nodes to fit the new records.
It re-organizes the nodes in the case of delete.
Good space utilization nodes contain only to the pointers to
the records and leaf nodes contain records.
It is suitable for partial and range search too.

20. Disadvantages:
There is no rearrangement of nodes while insertion or
deletion, then it would be an overhead.
It takes little effort, time and space.
But this disadvantage can be ignored compared to the speed
of traversal.