Introduction of B+ Tree (original) (raw)

Last Updated : 18 Apr, 2026

A B+ Tree is an advanced data structure used in database systems and file systems to maintain sorted data for fast retrieval, especially from disk. It is an extended version of the B Tree, where all actual data is stored only in the leaf nodes, while internal nodes contain only keys for navigation.

Components of B+ Tree

• Leaf nodes store all the key values and pointers to the actual data.
• Internal nodes store only the keys that guide searches.
• All leaf nodes are linked together, supporting efficient sequential and range queries.

This structure makes the B+ Tree balanced, disk-efficient, and ideal for database indexing.

Features of B+ Trees

• **Balanced: Auto-adjusts when data is added or removed, keeping search time efficient.
• **Multi-level: Has a root, internal nodes, and leaf nodes (which store the data).
• **Ordered: Maintains sorted keys, making range queries easy.
• **High Fan-out: Each node has many children, keeping the tree short and fast.
• **Cache-friendly: Works well with memory caches for better speed.
• **Disk-efficient: Ideal for disk storage due to fast data access.

How B+ Trees Work

The Structure of the Internal Nodes of a B+ Tree of Order 'a' is as Follows

• Each internal node is of the form: <P1, K1, P2, K2, ....., Pc-1, Kc-1, Pc> where c <= a and each **P i is a tree pointer (i.e points to another node of the tree) and, each **K i is a key-value (see diagram-I for reference).
• Every internal node has : K1 < K2 < .... < Kc-1
• For each search field value 'X' in the sub-tree pointed at by Pi, the following condition holds: Ki-1 < X <= Ki, for 1 < I < c and, Ki-1 < X, for i = c (See diagram I for reference)
• Each internal node has at most 'a' tree pointers.
• The root node has, at least two tree pointers, while the other internal nodes have at least \frac{a}{2} tree pointers each.
• If an internal node has 'c' pointers, c <= a, then it has 'c - 1' key values.

Structure of Internal Node

The Structure of the Leaf Nodes of a B+ Tree of Order 'b' is as Follows

• Each leaf node is of the form: <<K1, D1>, <K2, D2>, ....., <Kc-1, Dc-1>, Pnext> where c <= b and each **D i is a data pointer (i.e points to actual record in the disk whose key value is K i or to a disk file block containing that record) and, each **K i is a key value and, **P next points to next leaf node in the B+ tree (see diagram II for reference).
• Every leaf node has : K1 < K2 < .... < Kc-1, c <= b
• Each leaf node has at least \frac{b}{2} values.
• All leaf nodes are at the same level.

Structure of Leaf Node

In the below diagram, we can see that using the Pnext pointer it is viable to traverse all the leaf nodes, just like a linked list, thereby achieving ordered access to the records stored in the disk.

Tree Pointer

Searching a Record in B+ Trees

To search for a key:

1. Start at the root node
2. Navigate through internal nodes based on key comparisons
3. Reach the appropriate leaf node
4. If the key exists, return the data; otherwise, report "record not found"

**Example: In the image below, in order to search for 58, traverse the root → internal node → reach the correct leaf node that contains 58 (if it exists).

B+ tree

Insertion in B+ Trees

Insertion in B+ Trees is done in three steps:

1. Navigate to the correct leaf node
2. Insert the new key in sorted order

3. If overflow occurs:

• Split the leaf node
• Push the middle key to the parent
• This process may propagate up to the root

For more, refer to Insertion in a B+ Trees.

Deletion in B+Trees

Deletion involves:

1. Finding and removing the key from the appropriate leaf node

2. Rebalancing the tree to maintain B+ Tree properties:

• Borrowing keys from siblings or
• Merging nodes if necessary

For more, refer to Deletion in B+ Trees.

Difference Between B+ Tree and B Tree

Some differences between B+ Tree and B Tree are stated below.

Advantages

• A B+ tree with 'l' levels can store more entries in its internal nodes compared to a B-tree having the same 'l' levels. This accentuates the significant improvement made to the search time for any given key. Having lesser levels and the presence of Pnext pointers imply that the B+ trees is very quick and efficient in accessing records from disks.
• Data stored in a B+ tree can be accessed both sequentially and directly.
• It takes an equal number of disk accesses to fetch records.
• B+trees have redundant search keys, and storing search keys repeatedly is not possible.

Disadvantages

• **Slower exact match: Data is always in leaf nodes, increasing search path.
• **More disk access: Requires reaching the leaf even for a single record.
• **Complex updates: Insertion and deletion are more complex.
• **Extra space: Leaf node links need additional pointers.