Welcome to B-trees! A B-tree is a self-balancing tree data structure that's fundamental to modern database systems and file storage. Unlike binary trees, B-tree nodes can contain multiple keys and have multiple children. This example shows a B-tree where each node can hold up to 3 keys. The root node contains keys 10, 20, and 30, with four child nodes containing the remaining sorted values. This structure ensures all leaf nodes are at the same level, maintaining perfect balance.
B-trees are defined by their order, which determines the structure constraints. For a B-tree of order m, each node can have at most m children and m minus 1 keys. In this example with order 4, each node can have 2 to 4 children and 1 to 3 keys. The keys within each node are always sorted, and they act as separators that define the ranges for child subtrees. This ensures that searching follows a clear path down the tree.
Let's see how searching works in a B-tree. To find key 25, we start at the root and compare it with the node's keys: 10, 20, and 30. Since 25 is greater than 20 but less than 30, we follow the third pointer to the appropriate child node. In that child node, we find our target key 25. This process takes logarithmic time because we eliminate large portions of the tree at each step.
B-tree insertion maintains the balanced structure through node splitting. When inserting key 22 into a node that already contains 25 and 28, the node becomes overfull. Since our B-tree has order 4, nodes can hold at most 3 keys. The overflow triggers a split: the middle key 25 is promoted to the parent, while 22 and 28 become separate child nodes. This splitting process ensures the tree remains balanced and all operations maintain logarithmic time complexity.
B-trees are fundamental to modern computing systems. They're extensively used in database management systems like MySQL and PostgreSQL for indexing, enabling fast data retrieval. File systems such as NTFS and HFS+ use B-trees to organize directory structures efficiently. The key advantage is minimizing expensive disk I/O operations by storing multiple keys per node, matching the block size of storage devices. This makes B-trees ideal for systems where data doesn't fit entirely in memory, providing guaranteed logarithmic performance for all operations.