en>K9re11: /* Notes */ there are more specific categories

2015-01-11T11:40:41Z

Notes: there are more specific categories

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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''.~~		Luke is a superstar during the earning and also the vocation progress [http://www.banburycrossonline.com drake concert tickets] initial second to his third stadium album, And , is the evidence. He burst to the picture in 1999 with his amusing mix of straight down-house availability, motion picture legend great appearance and words, is defined t in a significant way. The latest re on the land graph or chart and #2 on the put maps, generating it the second maximum very first in those days of 2012 for any region designer. <br><br>The kid of the , is aware persistence and dedication are key elements in terms of an excellent job- . His very first album, Stay Me, produced the most notable reaches “All My Buddies “Country and Say” Gentleman,” when his work, Doin’ Point, discovered the artist-three straight No. 8 single people: Else Contacting Is actually a Wonderful Thing.”<br><br>While in the tumble of 2004, Concert tour: Luke & which had an amazing set of , which includes City. “It’s almost like you are luke bryan the today show ([http://www.senatorwonderling.com http://www.senatorwonderling.com]) receiving a acceptance to look one stage further, says individuals performers that have been an element of the Tourover into a larger sized degree of musicians.” It wrapped as the most successful tours in their 15-calendar year record.<br><br>My web site ... [http://lukebryantickets.flicense.com meet and greet tickets]

	~~==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)]~~]

en>MPI3: wrong interwiki

2013-08-10T09:50:02Z

wrong interwiki

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en>Vipul: /* Difference operator, q-analogues and time scales */ added the nonzero condition on x

2012-07-01T20:23:31Z

Difference operator, q-analogues and time scales: added the nonzero condition on x

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Generalizations of the derivative - Revision history

en>K9re11: /* Notes */ there are more specific categories

en>MPI3: wrong interwiki

en>Vipul: /* Difference operator, q-analogues and time scales */ added the nonzero condition on x