1. B+-Tree Deletion Underflow conditions B+ tree Deletion Algorithm
Leaf key rotations
Deletion Summary
Deletion Case 1: No underflow
Deletion Case 2: Key borrowing (key rotation) from adjacent leaf sibling
Deletion Case 3: Leaf merging
Deletion Case 4: Internal key borrowing (internal key rotation)
Case 5: Merging internal nodes

2. Deletion in B+ Tree: Underflow conditions
Like insertion, deletion must be on a leaf node.
A B+ tree has two UNDEFLOW conditions:
Leaf underflow Condition:
A leaf underflows if after deleting a key from it, it contains
L/2 - 1 keys
Internal node underflow Condition:
An internal node (excluding the root node) underflows if in the key deletion process it contains M/2 - 2 keys

3. B+ Tree Deletion Algorithm
To delete a key targetKey (and its associated record), we search for it. If it is not found in a leaf we report an error. If it is found at a leaf, say x, we remove it and its data reference. There are two issues to handle:
First issue:
targetKey appears as a separating key in some internal node
targetKey can appear in at most one ancestor y of x as a separating key. Moreover, we must have visited y and seen targetKey in it when we searched down the tree. So after deleting targetKey from x, we access y and replace targetKey by a copy of the new smallest key in node x.
After handling the first issue, we handle the second issue:
Second issue:
If after deleting targetKey, there is no leaf underflow, the deletion is complete; otherwise, if there is a leaf underflow (After deleting targetKey, node x contains L/2 - 1 keys) then:
If there is an adjacent leaf sibling with at least  L/2  + 1 keys we borrow from the
sibling the minimum key (if right sibling) or the maximum key (if left sibling). If no
adjacent sibling leaf with at least  L/2  + 1 keys exists, then we have to merge two
leaves. 3

4. Deletion in B+ Tree: Leaf key rotation
Let u be the node with leaf underflow.
Leaf left key rotation (borrowing from adjacent right sibling v):
Move the minimum key of v to u
Replace the separating key between u and v with a copy of the new minimum in v 4

5. Deletion in B+ Tree: Leaf key rotation (cont’d)
Let u be the node with leaf underflow.
Leaf right key rotation (borrowing from adjacent left sibling v)
Move the maximum key of v to u
Replace the separating key between u and v with a copy of the new minimum in u 5

6. Deletion Summary
We first fix the separator issue.
After that, we check if the leaf underflows. If it is the case, we always try to borrow from an adjacent sibling (first looking at adjacent right sibling, then, if not possible, adjacent left sibling).
If an adjacent sibling x has some extra keys, we ‘borrow’ an extra key from the sibling (min if x is a right sibling and max if it is a left one), COPY the new minimum key of the right sibling to the separating key in the parent, and fix the references DONE
Else, if ALL the siblings have the minimum number of keys L/2, we need to merge the underflow leaf with an adjacent sibling (right sibling, then, if not possible, left sibling).
Let u be the leaf node with underflow. Let v be the adjacent sibling:
Move the keys in u to v.
Remove the reference to u at parent [i.e., delete u]
Delete the separating key between u and v from the parent.
The merge process deletes one key from the parent, so the parent may underflow  need to continue the process  in the worst case, we may go up to the root:
If an internal node underflows use the borrowing or merge algorithms given in
slide 15, 16 and 19 6

7. Deletion Summary (cont’d)
Right leaf merging: 7

8. Deletion Summary (cont’d)
Left leaf merging: 8

9. Deletion in B+ Tree - Case1: No underflow
Example: Delete 20 from the following B+ tree of order M = 3 and L = 3
Delete 20
No leaf underflow

10. Deletion in B+ Tree- Case 2a: Leaf key borrowing from Right sibling
Example: Delete 25 from the following B+ tree of order M = 3 and L = 3
Delete 25
Leaf underflow
Borrow min key 30 from right sibling
Replace 30 in parent by a copy of new minimum in right sibling 10

11. Deletion in B+ Tree – Case 2b: Leaf key borrowing from left sibling
Example: Delete 17 from the following B+ tree of order M = 3 and L = 3
Delete 17
Leaf underflow
Borrow max key 13 from left sibling
Replace 15 in parent by a copy of new minimum in right sibling 11

12. Deletion in B+ Tree – Case 3a: Right Leaf Merging
Example: Delete 15 from the following B+ tree of order M = 4 and L = 3
Delete 15
Leaf underflow
Cannot borrow. Merge overflow node with adjacent right sibling 12

13. Deletion in B+ Tree – Case 3b: Left Leaf Merging
Example: Delete 19 from the following B+ tree of order M = 4 and L = 3
Delete 19
Leaf underflow
Cannot borrow. Merge overflow node with adjacent left sibling 13

14. Deletion in B+ Tree – Case 4: Internal Key Borrowing
An internal node u underflows:
Case 4a [Internal Left key rotation] : the adjacent right sibling v of u has at least  M/2 keys.
Move the separating key between u and v in the parent of u and v down to u.
Make the leftmost child of v the rightmost child of u.
MOVE the leftmost key in v to become the separating key between u and v in the parent of u and v.
Note: Contrast with Leaf key borrowing where the leftmost key in the right sibling is COPIED up to the parent to replace the separating key 14

15. Deletion in B+ Tree: Internal Key Borrowing (cont’d)
An internal node u underflows:
Case 4b [Internal Right key rotation] : the adjacent left sibling v of u has at least  M/2 keys.
Move the separating key between u and v in the parent of u and v down to u.
Make the rightmost child of v the leftmost child of u.
MOVE the leftmost key in u to become the separating key between u and v in the parent of u and v.
Note: Contrast with Leaf key borrowing where the leftmost key in the right sibling is COPIED up to the parent to replace the separating key 15

16. Internal node underflow
Deletion in B+ Tree - Example of Borrowing internal key from right sibling
An internal node u underflows:
Case 4a : the adjacent right sibling of v of u has at least  M/2 keys.
Example: Delete 26 from the following B+ tree of order M = 5 and L = 4
Delete 26
Leaf underflow
Cannot borrow. Merge overflow node with adjacent right sibling
Borrow min key 45 from adjacent right sibling
Internal node underflow 16

17. Deletion in B+ Tree – Example of Borrowing internal key from right sibling (cont’d)17

18. Internal node underflow
Deletion in B+ Tree – Example of Borrowing internal key from left sibling
An internal node u underflows:
Case 4b : the left sibling of v of u has at least  M/2 keys
Example: Delete 20 from the following B+ tree of order M = 5 and L = 4
Delete 20
Leaf underflow
Cannot borrow. Merge overflow node with adjacent left sibling
Borrow max key 17 from adjacent left sibling
Internal node underflow 18

19. Deletion in B+ Tree – Example of Borrowing internal key from left sibling (cont’d)19

20. Deletion in B+ Tree – Case 5: Merging internal nodes
An internal node w underflows:
Case 5 [Merging] : each of the adjacent right and left sibling of w has  M/2 - 1 keys.
Let v be one of the siblings
Merge w and v.
Move the separating key between w and v in the parent of w and v down to w. Note that this corresponds to deleting separating key from the parent of w and v.
Move the keys and child references in w to v.
Remove the reference to w in the parent.
merge node, adjacent right sibling and the
separating key x
If the parent of the merged node underflows, the merging process propagates upward. In the limit, a root with one key is deleted and the height decreases by one. 20

21. Deletion in B+ Tree – Case 5: Merging internal nodes (cont’d)
Note: The merging could also be done by using the adjacent left sibling instead of the adjacent right sibling.
merge node, adjacent left sibling and the
separating key v 21

22. Deletion in B+ Tree – Case5: Merging internal nodes (cont’d)
An internal node u underflows:
Case 5 : each of the right and left sibling of u has  M/2 - 1 keys.
Example: Delete 34 from the following B+ tree of order M = 5 and L = 4
Delete 34
Leaf underflow
Cannot borrow. Merge overflow node with adjacent left sibling
Cannot borrow. Merge node with adjacent right sibling
Internal node underflow 22

23. Deletion in B+ Tree – Case5: Merging internal nodes (cont’d)23

24. Deletion in B+ Tree – Case 5: Merging the root
Example: Delete 20 from the following B+ tree of order M = 3 and L = 3
Delete 20
underflow
Cannot borrow. Merge overflow node with adjacent left sibling 24