Generalizations of the derivative

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 height h of a k-ary tree does not include the root node, with a tree containing only a root node having a height of 0.
• The total number of nodes in a perfect k-ary tree is $(k^{h+1}-1)/(k-1)$, while the height h is

$\lceil {\log }_k(k-1)+{\log }_k(\mathit{n}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\_\mathit{o}\mathit{f}\_\mathit{n}\mathit{o}\mathit{d}\mathit{e}\mathit{s})-1\rceil .$

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

Note : Formula is not applicable for a 2-ary tree with height 0, as the ceiling operator approximates and simplifies the true formula, which can be described as

${\log }_k[(k-1)*\mathit{n}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\_\mathit{o}\mathit{f}\_\mathit{n}\mathit{o}\mathit{d}\mathit{e}\mathit{s}+1]-1,k\ge 2.$

Methods for storing k-ary trees

Arrays

k-ary trees can also be stored in breadth-first order as an implicit data structure in arrays, and if the tree is a complete k-ary tree, this method wastes no space. In this compact arrangement, if a node has an index i, its c-th child is found at index $k*i+1+c$, while its parent (if any) is found at index $\lfloor \frac{i-1}{k}\rfloor$ (assuming the root has index zero). This method benefits from more compact storage and better locality of reference, particularly during a preorder traversal.

See also

• Left-child right-sibling binary tree
• Binary tree

References

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External links

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