Generalizations of the derivative: Difference between revisions

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In [[graph theory]], a '''k-ary tree''' is a rooted [[tree (graph theory)|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''.
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==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 <ref name="orderedtrees">{{cite web|url=http://cs.lmu.edu/~ray/notes/orderedtrees/|title=Ordered Trees|accessdate=19 November 2012}}</ref> k-ary tree in which all [[leaf node]]s are at the same depth.<ref>{{cite web|url=http://xlinux.nist.gov/dads/HTML/perfectKaryTree.html|title=perfect k-ary tree|last=Black|first=Paul E.|date=20 April 2011|publisher=U.S. National Institute of Standards and Technology|accessdate=10 October 2011}}</ref>
*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". <ref name="orderedtrees"></ref>
 
==Properties of k-ary trees==
 
*For a k-ary tree with height ''h'', the upper bound for the maximum number of leaves is <math>k^h</math>.
* 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 <math>(k^{h+1} - 1)/(k-1)</math>, while the height ''h'' is
 
:<math>\left\lceil\log_k (k - 1) + \log_k (\mathit{number\_of\_nodes}) - 1\right\rceil.</math>
 
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
:<math>\log_k [(k - 1) * \mathit{number\_of\_nodes} + 1] - 1, k \ge 2.</math>
 
==Methods for storing k-ary trees==
 
===Arrays===
 
k-ary trees can also be stored in breadth-first order as an [[implicit data structure]] in [[array data structure|array]]s, 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 <math>k*i + 1 + c</math>, while its parent (if any) is found at index ''<math>\left \lfloor \frac{i-1}{k} \right \rfloor</math>'' (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==
{{Reflist}}
*{{cite book
   |last = Storer
  |first = James A.
  |title = An  Introduction to Data Structures and Algorithms
  |year = 2001
  |isbn = 3-7643-4253-6
  |publisher = Birkhäuser Boston}}
 
==External links==
*[http://www.brpreiss.com/books/opus5/html/page257.html N-ary trees, Bruno R. Preiss, P.Eng.]
 
[[Category:Trees (graph theory)]]

Latest revision as of 12:40, 11 January 2015

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