What is a B-tree and where is it used?

A B-tree is a self-balancing search tree generalization of BST where each node can have multiple keys and multiple children (between t and 2t children for minimum degree t). Properties: (1) All leaves are at the same depth (balanced); (2) Each node (except root) has at least ⌈m/2⌉-1 keys (at least half full); (3) Each node has at most m-1 keys (m children) — m is the order; (4) Keys within a node are sorted; (5) A node with k keys has k+1 children. Operations: O(log n) for search, insert, delete. Why B-trees for databases/filesystems: designed to minimize disk I/O — nodes correspond to disk pages (typically 4KB-16KB); wide branching factor means very few disk accesses for any lookup (a B-tree with billions of entries has height ~3-4). B+ tree (most databases, including MySQL InnoDB, PostgreSQL): all actual data (values) stored in leaf nodes; internal nodes store only keys for routing; leaf nodes linked as a doubly linked list — enables efficient range scans (walk from first matching key through linked leaf list). B- vs B+: B+ tree requires range scans in O(k+log n) vs O(k log n) for B-tree. Applications: (1) Database indexes (primary key, secondary indexes); (2) File systems (HFS+, NTFS, ext4, Btrfs); (3) Key-value stores; (4) In-memory databases.