This document discusses B-trees and algorithms for inserting and deleting keys from a B-tree. It explains that B-TREE-INSERT uses B-TREE-SPLIT-CHILD to guarantee recursion never descends to a full node. When the root becomes full during an insertion, it is split and a new root node is created, increasing the tree height by one. For deletion, it ensures a node does not become too small by potentially redistributing keys between nodes. Examples are provided to illustrate deleting keys from B-trees.

1. File Processing & Organization
Course No: 5901227-3
Lecture 7
B-Trees (Continued)

2. Inserting a key into a B-tree in a single
pass down the tree
• The B-TREE-INSERT procedure uses B-TREE-SPLIT-CHILD to
guarantee that the recursion never descends to a full node.
B-TREE-INSERT(T, k)
1 r ← root[T]
2 if n[r] = 2t - 1
3 then s ← ALLOCATE-NODE()
4 root[T] ← s
5 leaf[s] ← FALSE
6 n[s] ← 0
7 c1[s] ← r
8 B-TREE-SPLIT-CHILD(s, 1, r)
9 B-TREE-INSERT-NONFULL(s, k)
10 else B-TREE-INSERT-NONFULL(r, k)

3. • Lines 3-9 handle the case in which the root node r
is full: the root is split and a new node s (having
two children) becomes the root. Splitting the root
is the only way to increase the height of a B-tree.
Figure 1 illustrates this case.
• The procedure finishes by calling BTREE-INSERT-
NONFULL to perform the insertion of key k in the
tree rooted at the nonfull root node.

4. Figure 1
• Splitting the root with t = 4. Root node r is split in two, and a
new root node s is created. The new root contains the median
key of r and has the two halves of r as children.
• The B-tree grows in height by one when the root is split.

5. B-TREE-INSERT-NONFULL
• B-TREE-INSERT-NONFULL(x, k)
1 i ← n[x]
2 if leaf[x]
3 then while i ≥ 1 and k < keyi[x]
4 do keyi+1[x] ← keyi[x]
5 i ← i - 1
6 keyi+1[x] ← k
7 n[x] ← n[x] + 1
8 DISK-WRITE(x)
9 else while i ≥ 1 and k < keyi[x]

6. 10 do i ← i - 1
11 i ← i + 1
12 DISK-READ(ci[x])
13 if n[ci[x]] = 2t - 1
14 then B-TREE-SPLIT-CHILD(x, i, ci[x])
15 if k> keyi[x]
16 then i ← i + 1
17 B-TREE-INSERT-NONFULL(ci[x], k)

7. • Lines 3-8 handle the case in which x is a leaf node
by inserting key k into x. If x is not a leaf node,
then we must insert k into the appropriate leaf
node in the subtree rooted at internal node x.
• In this case, lines 9-11 determine the child of x to
which the recursion descends.
• Lines 13-16 guarantee that the procedure never
recurses to a full node.
• Line 17 then recurses to insert k into the
appropriate subtree.

8. Deleting a key from a B-tree
• Just as we had to ensure that a node didn't
get too big due to insertion, we must ensure
that a node doesn't get too small during
deletion

9. Deleting a key from a B-tree
• Delete key B from a B-Tree with t=3.
D G
A B C E F H I

10. Deleting a key from a B-tree
We want to delete key F from a B-Tree
with t =3.
D G
A B C E F H I

11. Deleting a key from a B-tree
We want to delete key F from a B-Tree
with t =3.
G
A B C D E F H I

12. Deleting a key from a B-tree
We want to delete key F from a B-Tree
with t =3.
C G
A B D E F H I

13. Deleting a key from a B-tree
We want to delete key S from a B-Tree
with t=3.
Q T
OP R S U X

14. Deleting a key from a B-tree
We want to delete key S from a B-Tree
with t=3.
Q T
OP R S T U X

15. Deleting a key from a B-tree
Q U
O P R S T W V

16. Deleting a key from a B-tree
We want to delete key U from a B-Tree
with t=3.
Q T
O P R S W V

17. Deleting a key from a B-tree
• We want to delete I from a B-Tree with t =3.
I H
G M J K L O P

18. Deleting a key from a B-tree
• We want to delete I from a B-Tree with t =3.
J H
G M K L O P

19. Deleting a key from a B-tree
• We want to delete U from a B-Tree with t =3.
R U X
P Q ST V W Y Z

20. Deleting a key from a B-tree
• We want to delete U from a B-Tree with t =3.
R X
P Q S T V W Y Z