K-ary tree

In graph theory, a k-ary tree is a rooted tree in which each node has no more than k children. It is also sometimes known as a k-way tree, an N-ary tree, or an M-ary tree. A binary tree is the special case where k=2.

Types of k-ary trees

• A full k-ary tree is a k-ary tree where within each level every node has either 0 or k children.
• A perfect k-ary tree is a full k-ary tree in which all leaf nodes are at the same depth.
• A complete k-ary tree is a k-ary tree which is maximally space efficient. It must be completely filled on every level (meaning that each level has k children) except for the last level (which can have at most k children). However, if the last level is not complete, then all nodes of the tree must be "as far left as possible".

Properties of k-ary trees

• For a k-ary tree with height h, the upper bound for the maximum number of leaves is $k^h$.
• The total number of nodes is $\frac{k^{h + 1} - 1}{k - 1}$, while the height h is

$\left\lceil\log_k (k - 1) + \log_k (\mathit{number\_of\_nodes}) - 1\right\rceil.$

Note : A Tree containing only one node is taken to be of height 0 for this formula to be applicable.

Note : Formula applicable only for number_of_nodes = number of nodes in *complete* k-ary tree

References

• Storer, James A. (2001). An Introduction to Data Structures and Algorithms. Birkhäuser Boston. ISBN 3-7643-4253-6. 

External links

• N-ary trees, Bruno R. Preiss, P.Eng.