The document discusses B-trees, which are self-balancing tree data structures that allow efficient insertion and deletion of data while keeping the tree height shallow. B-trees allow for efficient searching, insertion, and deletion of data in logarithmic time by allowing nodes to have more than two child nodes, and by splitting and merging nodes as needed to balance the tree during operations. The document covers the basic structure and properties of B-trees and explains the algorithms for insertion and deletion of keys through cases involving splitting, merging, redistribution and underflow of nodes.

1. B-Tree
An Analysis
By:
Nikhil Sharma
BE/8034/09

2. Definition
A B-tree is a tree data structure that keeps data sorted and allows
searches, sequential access, insertions, and deletions in
logarithmic amortized time. The B-tree is a generalization of
a binary search tree in which more than two paths diverge from a
single node.
A B-tree of order m (the maximum number of children for each
node) is a tree which satisfies the following properties:
 Every node has at most m children.
 Every node (except root and leaves) has at least m⁄2 children.
 The root has at least two children if it is not a leaf node.
 All leaves appear in the same level, and carry information.
 A non-leaf node with k children contains k−1 keys.
Declaration in C:
typedef struct { int Count; // number of keys stored in the current node
ItemType Key[3]; // array to hold the 3 keys [4];
long Branch[4]; // array of fake pointers (record numbers)
} NodeType;

3. Order & Key of a B-Tree
The following is an example of a B-tree of order 5. This means
that (other than the root node) all internal nodes have at least 3
children (and hence at least 2 keys). Of course, the maximum
number of children that a node can have is 5 (so that 4 is the
maximum number of keys). In practice B-trees usually have
orders a lot bigger than 5. The first row in each node shows the
keys, while the second row shows the pointers to the child nodes

4. Height of B-Tree
 If n ≥ 1, then for any n-key B-tree T of height h and
minimum degree t ≥ 2, h logt (n 1)/2
 Height of the B-Tree with n keys is important as it bound
the number of disk accesses.
 The height of the tree is maximum when each node has
minimum number of the subtree pointers, q m / 2
.
 Note:If number of nodes in B-tree equal 2,000,000 (2
million) and m=200 then maximum height of B-tree is
3, where as the binary tree would be of height 20.

5. Search in a B-Tree
 Search in a B-tree is similar to the search in BST except that in B-
tree we make a multiway branching decision instead of binary
branching in BST.
25 62
12 19 32 39 73 84
3 5 15 17 21 23 30 31 34 37 45 51 69 71 75 79 90 94
Search key 71

6. B-Tree Insert Operation
 Insertion in B-tree is more complicated than in BST.
 In BST, the keys are added in top down fashion
resulting in an unbalanced tree.
 B-tree is built bottom up, the keys are added in the
leaf node, if the leaf node is full another node is
created, keys are evenly distributed and middle key
is promoted to the parent. If parent is full, the
process is repeated.
 B-tree can also be built in top down fashion using
pre-splitting technique.

7. Basic Idea : Insertion
Find position for the key
in the appropriate leaf node
Insert key in order Is node
and adjust pointer
No full ?
yes
Split node: If parent is full
• Create a new node
• Move half of the keys from the full node to
the new node and adjust pointers
• Promote the median key (before split)
to the parent
Split guarantees that each node has m/ 2 1
keys.

8. Cases in B-Tree Insert Operation
 In B-tree insertion we have the following
cases:
◦ Case 1: The leaf node has room for the new
key.
◦ Case 2: The leaf in which key is to be placed is
full.
 This case can lead to the increase in tree height.

9. B-Tree Insert Operation
 Case 1: The leaf node has room for the new key.
Find appropriate leaf
Insert 3 node for key 3
3
10 25
5 8 14 19 20 23 32 38
Insert 3 in order

10. B-Tree Insert Operation
 Case 2: The leaf in which key is to be placed is full.
Find appropriate leaf
Insert 16 node for key 16
16
10 25
19
3 5 8 14 19 20 23 32 38
No room for key 16 in leaf node
Insert key 19 in parent node in order
Move median key 19 up and
Split node: create a new node
and move keys to the new
node.
14 16 20 23
19

11. B-Tree Insert Operation
 Case 2: The leaf in which key is to be placed is full
and this lead to the increase in tree height.
45 55 67 81

12. B-Tree Insert Operation
 Case 2: The height of the tree increases.
Insert 16
No room for 27 in parent, Split node
Insert 27 in parent in order 55
16
45 55 67 81
55
No room for 19 in parent,
Split parent node 48 52 57 61 72 77 86 92
13 27
19
27 33 38
3 3 4 5 5 7 3 3 4 5 5 7 3 3 4 5 5 7 3 3 4 5 5 7
2 8 7 1 9 5 2 8 7 1 9 5 2 8 7 1 9 5 2 8 7 1 9 5
9 12 14 19 20 23 29 31 35 36 41 42
Insert 19 in parent node in order
No room for key 16,
Move median key 19 up & Split node
19
14 16 20 23

13. B-Tree Delete Operation
 Deletion is analogous to insertion, but a
little more complicated.
 Two major cases
◦ Case 1: Deletion from leaf node
◦ Case 2: Deletion from non-leaf node
 Apply delete by copy technique used in BST, this
will reduce this case to case 1.
 In delete by copy, the key to be deleted is
replaced by the largest key in the right subtree or
smallest in left subtree (which is always a leaf).

14. B-Tree Delete Operation
 Leaf node deletion cases:
◦ After deletion node is at least half full.
◦ After deletion underflow occurs
 Redistribute: if number of keys in siblings > 2 .
m
1
 Merge nodes if number of keys in siblings < m 2
1 .
 Merging leads to decrease in tree height.

15. B-Tree Delete Operation
 After deletion node is at least half full. (inverse of insertion
case 1)
Search key 3
10 25
3 5 8 14 19 32 38 40 45
Key found, delete key 3.
Move others keys in the node to eliminate
the gap.

16. B-Tree Delete Operation
 Underflow occurs, evenly redistribute the keys if left or right
sibling has keys .
m/ 2 1
Search key
Delete 14
14
10 25
5 8 14 19 32 38 40 45
Underflow occurs, evenly redistribute
keys
in the underflow node, in its sibling
and the separator key.

17. B-Tree Delete Operation
 Underflow occurs and the keys in the left & right sibling are
m / 2 1 . Merge the underflow node and a sibling.
Delete 25 Move separator key down.
Move the keys to underflow
10 32
node and discard the sibling.
5 8 19 25 38 40
Underflow occurs, merge
nodes.

18. B-Tree Delete Operation
 Underflow occurs, height decreases after merging.
Delete 21
70
Underflow
occurs, merge
nodes
8 32 79 85
3 5 21 27 47 66 73 75 78 81 83 88 90 92
Underflow occurs, merge
nodes by moving separator
key and the keys in sibling
node to the underflow
node.

19. B-Tree V/s Binary Tree
Advantages
 Efficient in real life problems where
number of records is very large (i.e.
large datasets)
 Frees up RAM as all nodes located
on secondary memory
 B Tree reduces depth of the tree
hence, desired record is located
faster
Disadvantages
 Decision process at each node is
more complicated in a B-tree
 A sophisticated program is required
to execute the operations in a B-tree
Fig. Comparison of linear growth rate vs.
logarithmic growth rate

20. The End