What is B-tree? Construct a B-tree of order 3 for the following set of input data: 69, 43, 25, 40, 16, 100, 124, 7, 15, 8

Q.) What is B-tree? Construct a B-tree of order 3 for the following set of input data: 69, 43, 25, 40, 16, 100, 124, 7, 15, 8
Subject: Data Structure and Algorithm

B-tree

A B-tree is a self-balancing tree data structure that maintains data sorted and allows efficient searches, insertions, and deletions. It is commonly used in database systems and file systems.

Properties of a B-tree:

Order: The order of a B-tree, denoted by m, is the maximum number of children a node can have. A B-tree of order m is also known as an m-ary B-tree.
Root node: The root node of a B-tree is always at least half full, except for the case when it is the only node in the tree.
Internal nodes: Internal nodes of a B-tree can have a maximum of m children and a minimum of ⌈m/2⌉ children, except for the root node.
Leaf nodes: Leaf nodes of a B-tree contain the actual data items and have a maximum of m children and a minimum of ⌈m/2⌉ children.
Height: The height of a B-tree is the maximum number of levels from the root node to any leaf node.

Searching in a B-tree:

To search for a data item in a B-tree, the following steps are taken:

Start from the root node.
If the data item is found in the current node, return it.
If the data item is not found in the current node, choose the appropriate child node to continue the search and go to Step 2.

Inserting into a B-tree:

To insert a new data item into a B-tree, the following steps are taken:

Start from the root node.
If the current node has space for a new child, insert the new data item into the node and maintain the sorted order.
If the current node is full, split it into two nodes and redistribute the data items among them.
Update the parent node to reflect the changes.

Deleting from a B-tree:

To delete a data item from a B-tree, the following steps are taken:

Start from the root node.
If the data item is found in the current node, delete it from the node and maintain the sorted order.
If the data item is not found in the current node, choose the appropriate child node to continue the search and go to Step 2.
If a node becomes underfull after deletion (i.e., it has fewer than ⌈m/2⌉ children), merge it with a neighboring node or redistribute data items from a neighboring node.
Update the parent node to reflect the changes.

Constructing a B-tree of Order 3

Given the set of input data:

69, 43, 25, 40, 16, 100, 124, 7, 15, 8

We can construct a B-tree of order 3 as follows:

            69
          /    \
        40      100
       / \     /  \
     25  43  7    124
    / \    / \
   16 8  15

The root node contains the value 69. The left child of the root node contains the values 25, 40, and 43. The right child of the root node contains the values 100 and 124. The left child of the node containing 40 contains the values 16 and 8. The right child of the node containing 40 contains the value 15.

This B-tree satisfies all the properties of a B-tree of order 3.