Select Page

# B+ tree Index Files

A B+ tree is a balanced binary search tree that follows a multi-level index format. The leaf nodes of a B+ tree denote actual data pointers. B+ tree ensures that all leaf nodes remain at the same height, thus balanced. Additionally, the leaf nodes are linked using a link list; therefore, a B+ tree can support random access as well as sequential access.

### Structure of B+ Tree

Every leaf node is at equal distance from the root node. A B+ tree is of the order n where nis fixed for every B+ tree.

Internal nodes

• Internal (non-leaf) nodes contain at least ⌈n/2⌉ pointers, except the root node.
• At most, an internal node can contain n pointers.

Leaf nodes

• Leaf nodes contain at least ⌈n/2⌉ record pointers and ⌈n/2⌉ key values.
• At most, a leaf node can contain n record pointers and n key values.
• Every leaf node contains one block pointer P to point to next leaf node and forms a linked list.

### B+ Tree Insertion

• B+ trees are filled from bottom and each entry is done at the leaf node.
• If a leaf node overflows −
• Split node into two parts.
• Partition at i = ⌊(m+1)/2⌋.
• First i entries are stored in one node.
• Rest of the entries (i+1 onwards) are moved to a new node.
• ith key is duplicated at the parent of the leaf.
• If a non-leaf node overflows −
• Split node into two parts.
• Partition the node at i = ⌈(m+1)/2.
• Entries up to i are kept in one node.
• Rest of the entries are moved to a new node.

### B+ Tree Deletion

• B+ tree entries are deleted at the leaf nodes.
• The target entry is searched and deleted.
• If it is an internal node, delete and replace with the entry from the left position.
• After deletion, underflow is tested,
• If underflow occurs, distribute the entries from the nodes left to it.
• If distribution is not possible from left, then
• Distribute from the nodes right to it.
• If distribution is not possible from left or from right, then
• Merge the node with left and right to it.