# (a,b)-tree

In computer science, an (a,b) tree is a kind of balanced search tree.

An (a,b)-tree has all of its leaves at the same depth, and all internal nodes except for the root have between a and b children, where a and b are integers such that 2 ≤ a ≤ (b+1)/2. The root has, if it is not a leaf, between 2 and b children.

## Definition

Let a, b be positive integers such that 2 ≤ a ≤ (b+1)/2. Then a rooted tree T is an (a,b)-tree when:

• Every inner node except the root has at least a and at most b children.
• The root has at most b children.
• All paths from the root to the leaves are of the same length.

## Internal node representation

Every internal node v of a (a,b)-tree T has the following representation:

• Let ${\displaystyle \rho _{v}}$ be the number of child nodes of node v.
• Let ${\displaystyle S_{v}[1\dots \rho _{v}]}$ be pointers to child nodes.
• Let ${\displaystyle H_{v}[1\dots \rho _{v}-1]}$ be an array of keys such that ${\displaystyle H_{v}[i]}$ equals the largest key in the subtree pointed to by ${\displaystyle S_{v}[i]}$.