What is a B-tree? Construct a B-tree of order 3 for the following set of input data: 69, 43, 25, 40, 12, 100, 14, 157, 8, 132

Understanding B-trees

A B-tree is a balanced search tree that is specifically designed for storing and efficiently searching large amounts of data on disk. It is commonly used in database systems and file systems due to its ability to handle large volumes of data and maintain efficient performance. B-trees are characterized by their order, which determines the number of child nodes each parent node can have.

Constructing a B-tree of Order 3

To construct a B-tree of order 3 for the given set of input data, we follow these steps:

1. Initialize the Root Node:

   • Create the root node of the B-tree.
   • Since the order of the B-tree is 3, the root node can have a maximum of 3 child nodes.

2. Insert Data into the Root Node:

   • Start inserting the data values into the root node in ascending order.
   • If the root node becomes full (has 3 child nodes), we split it into two nodes and promote the middle value to the parent node.

3. Repeat the Insertion Process:

   • Continue inserting the remaining data values into the B-tree by following these steps:
     • Find the appropriate leaf node to insert the new data value.
     • If the leaf node is not full (has fewer than 3 child nodes), insert the new data value into the node.
     • If the leaf node is full, split it into two nodes and promote the middle value to the parent node.

4. Balance the Tree:

• Ensure that the tree remains balanced after each insertion.
• If a node becomes unbalanced (has more or fewer child nodes than allowed), perform rotations or redistributions to maintain balance.

1. Repeat Steps 2-4 until All Data is Inserted:

• Continue inserting the remaining data values and balancing the tree until all data is successfully inserted into the B-tree.

Example: Constructing a B-tree of Order 3 for the Given Data

Consider the input data: 69, 43, 25, 40, 12, 100, 14, 157, 8, 132.

1. Initialize the Root Node:

   • Create the root node. Since the order is 3, it can have a maximum of 3 child nodes.

2. Insert Data into the Root Node:

   • Insert the data values in ascending order: 8, 12, 14, 25, 40, 43, 69, 100, 132, 157.
   • The root node becomes full, so we split it into two nodes: [8, 12] and [14, 25, 40].
   • Promote the middle value, 25, to the parent node.

3. Repeat the Insertion Process:

   • Continue inserting the remaining data values: 43, 69, 100, 132, 157.
   • Split nodes and promote middle values as necessary to maintain balance.

4. Balance the Tree:

   • Ensure that the tree remains balanced after each insertion.
   • Perform rotations or redistributions to maintain balance.

After completing the insertion and balancing process, the resulting B-tree of order 3 is:

              [8, 12, 25]
             /         \
           [4, 6]     [14, 20]
          /     \     /      \
        [1, 3]  [5, 7] [16, 18] [22, 24]
                      /         \
                    [21, 23]    [26, 28]
                               /
                             [29, 31]

This B-tree efficiently stores and organizes the given data, allowing for efficient searching and retrieval of data. The order of the B-tree (in this case, 3) determines the branching factor and the maximum number of child nodes per node, which impacts the performance and efficiency of the tree.