Introduction of B-Tree

A B-tree is a self-balancing tree data structure that maintains sorted data and allows for efficient insertion, deletion, and search operations. It is widely used in databases and file systems for indexing large volumes of data because of its ability to handle large datasets efficiently.

Key Characteristics of B-Trees:

1. Balanced Structure:
   • A B-tree remains balanced by ensuring that every leaf node is at the same level. This helps in keeping the search time logarithmic in relation to the number of elements in the tree.
2. Nodes with Multiple Keys:
   • Each node in a B-tree can store multiple keys and have multiple children, unlike binary trees which only store one key per node and have two children per node.
   • Each internal node contains keys that act as separation values, directing the search for specific data.
3. Order of B-Tree:
   • The order of a B-tree (often denoted by m) determines the maximum number of children each node can have.
   • For example, in a B-tree of order m, each node can have at most m children and m-1 keys.
4. Efficiency:
   • Search: O(log n)
   • Insertion: O(log n)
   • Deletion: O(log n)
   • This efficiency is achieved because of the way the tree is structured, where each level of the tree can contain a large number of keys and thus reduces the number of levels compared to a binary search tree.
5. Usage in Databases:
   • B-trees are often used in databases to create indexes because they allow for fast access to data while keeping the database balanced. The keys in a B-tree index allow for fast searching, insertion, and deletion, even as the database grows in size.

Operations:

• Search: Similar to binary search, but with more than two branches to follow.
• Insert: If a node overflows (i.e., the number of keys exceeds the maximum), it splits into two nodes, and the middle key is pushed up into the parent node.
• Delete: If a key is deleted and a node becomes underfilled, it may borrow a key from a neighboring sibling or merge with a sibling to maintain balance.

Example of B-Tree (Order 3):

                [10, 20]
               /    |    \
          [5]   [15]   [25, 30, 35]

In this example:

• The root node has two keys (10 and 20) and three children.
• The leftmost child holds keys less than 10, the middle one holds keys between 10 and 20, and the rightmost holds keys greater than 20.

The B-tree is highly efficient for read-heavy applications, such as databases or file systems, where large volumes of sorted data need to be accessed quickly.