Height-balanced tree

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	tree4.doc
		Height Balanced Trees A height balanced tree is one in which the difference in the height of the two subtrees for any node is less than or equal to some specified amount.
		Height balancing for a single node is still O(1) However, it would be expected that, on the average, only about half the insertions result in an unbalanced tree.
		Restoring the balance of the pivot node restores the balance of the whole sub-tree and potentially all of the nodes that were affected by the insertion.
	pages.cpsc.ucalgary.ca /~marina/331/tree4.doc (1940 words)

	AVL tree - Wikipedia, the free encyclopedia
		In an AVL tree the heights of the two child subtrees of any node differ by at most one, therefore it is also known as height-balanced.
		Insertion into an AVL tree may be carried out by inserting the given value into the tree as if it were an unbalanced binary search tree, and then retracing one's steps toward the root, rotating about any nodes which have become unbalanced during the insertion (see tree rotation).
		A node with balance factor -2 or 2 is considered unbalanced and requires rebalancing the tree.
	en.wikipedia.org /wiki/AVL_tree (501 words)

	Threaded, height-balanced binary tree data structure - US Patent 5557786
		The height of a tree may also be defined as the length, in terms of numbers of links plus one from the root entry, of the longest path from the root entry to an entry which has no links, such as Entry C of the tree 40.
		The height of a tree is the number of levels of hierarchy among the entries of the tree.
		The sub-tree headed by Entry F has a height of one on its right side and a height of zero on the left side.
	www.patentstorm.us /patents/5557786.html (7855 words)

	trees-AVL.doc
		In the following tree the heights of the nodes are indicated below: Nodes 40, 75, 90 : Height 1 Nodes 60, 80 Height 2 Node 70 : Height 3.
		B.F. If BF for any node is 2 or -2, we say that the tree is unbalanced.
		Consider the following tree which became unbalanced after insertion of node 98.
	longwood.cs.ucf.edu /courses/cop3502/fall04/trees-AVL.doc (981 words)

	Other Types of Trees: 3.1 Binary Height Balanced Trees
		Height balanced trees are used in order to maximize the efficiency of the operations on a tree.
		In Figure 1.1a, the left subtree of the tree has a depth of three, where the right subtree has a depth of two, so they are balanced, since their heights do not differ by greater than one.
		A binary height balanced tree is a binary tree that has equal depth throughout the tree, +/- one level.
	www.cs.usask.ca /resources/tutorials/csconcepts/1998_6/bintree/3-1.html (244 words)

	Algorithms Glossary
		Spanning tree of a graph G, subgraph which is a tree, contains all vertices of G, and whose edges form a subset of the edges of G. Sparse graph, one where E Sparse type, where the set of potential values is much bigger than the set of values actually in use, e.g.
		Complete binary tree, a full binary tree where all leaf nodes are at level `n' or `n-1' and all gaps, if any, are in level n at the right; also see perfect binary tree.
		Binary tree, a tree where each node has 0, 1 or 2 child nodes (subtrees); see strict binary tree.
	www.csse.monash.edu.au /~lloyd/tildeAlgDS/Glossary (1628 words)

	CS660: AVL Trees
		Height of a tree is the length of the longest path from the root to a leaf.
		A node has the HB[k] property if the height of the left and right subtrees of the node differ in height by at most k.
		Traversing Up from Left subtreeDeletion reduced the height of the left subtreeRight subtree was greater then the Left subtree Left subtrees has equal subsubtrees
	www.eli.sdsu.edu /courses/fall96/cs660/notes/avl/avl.html (542 words)

	TREE STRUCTURES
		A tree is height balanced if all its nodes have a BF of 1, 0, or -1.
		Height here is the number of nodes visited in traversing a branch that leads to a leaf node at the deepest level of the tree.
		The algorithm to height balance a binary tree (as insertions are being processed) is non-trivial and development costs can rise in trying to implement it.
	www.seasite.niu.edu /cs464/tree3.htm (988 words)

	Binary Trees
		Output is the tree of the expression that is the result of differentiation with respect to a given variable.
		It is difficult to print trees down the page, because they quickly grow too wide, but it is relatively easy to print them across the page so that the root is at the left and the tree grows to the right.
		For non-emptyTree trees, it may be appropriate to try a tree of a "few" nodes and a tree of a single node.
	www.csse.monash.edu.au /~lloyd/tildeAlgDS/Tree (2830 words)

	AVL Trees
		To maintain balance in a height balanced binary tree, each node will have to keep an additional piece of information that is needed to
		balance factor, and it indicates if the difference in height between the left and right subtrees is the same or, if not, which of the two subtrees has height one unit larger.
		Worst case height of an AVL tree with n Nodes
	www.academic.marist.edu /~jzbv/ads/AVLTrees.htm (839 words)

	Height Balanced Binary search trees
		An AVL tree has the same properties as a binary search tree, except that for any node in the tree, the height of the left and right subtrees can differ by no more than one level and the keys used to sort the data must be unique.
		The balance factor is calculated by subtracting the height of the right subtree from the height of the left subtree.
		Tree after node(19) is added an imbalance occurs at node(45) with bf=+2 and at node(9) with bf=-2.
	ironbark.bendigo.latrobe.edu.au /subjects/DS/mal/session200/lecture.html (542 words)

	B-Trees
		To ensure that the height of the tree is as small as possible and therefore provide the best running time, a balanced tree structure like a red-black tree, AVL tree, or b-tree must be used.
		Ideally, a tree will be balanced and the height will be log n where n is the number of nodes in the tree.
		B-trees are balanced trees that are optimized for situations when part or all of the tree must be maintained in secondary storage such as a magnetic disk.
	www.bluerwhite.org /btree (2387 words)

	No Title
		A height balanced leftist tree is a binary tree such that for every internal node, the s value (distance to fall off the tree) of the left child is greater than or equal to the s value of the right tree.
		A heap is a tree in which the root is larger in priority than its children.
		A leftist tree is a heap that allows efficent merging of priority queues.
	www.cs.usu.edu /~allanv/cs2200/Sahni9a/Sahni9a.html (210 words)

	AVL Trees
		As you move away from a balanced tree to a more linear structure, height approaches N and the maximum comparisons increases with the height of the the tree - ultimately becoming the same as searching a linked list.
		An AVL tree is a binary search tree in which the heights of the left and right subtrees of the root differ by at most 1, and in which the left and right subtrees are again AVL trees.
		With each node of an AVL tree is associated a balance factor which is left high, equal, or right high according, respectively, as to whether the left subtree has height greater than, equal to, or less than that of the right subtree.
	www.cs.wm.edu /~debbie/cs241/trees/AVLtrees.html (865 words)

	Lecture 13
		Red-black trees are binary search trees which come with additional information about the nodes: the nodes are coloured red or black.
		If after insertion the number becomes larger than 1 or less than -1 the tree is restructured.
		Instead we will study in detail the algorithm for AVL trees (which is bottom up).
	www.cs.nott.ac.uk /~nza/G5BADS00/l13.html (1009 words)

	CS261 - Trees Pt 1
		In a height balanced tree the absolute value of the difference at any node between the longest path in the left and right subtrees is at most one.
		If we let Mn represent the function yielding the minumum number of nodes for a height balanced tree of depth n, we obtain:
		A complete tree of height n has at least 2^n nodes.
	web.engr.oregonstate.edu /~dambrosi/cs261/trees/treesNotes.html (1610 words)

	Height Balanced Trees
		The reason it is important is that in a height-balanced tree, the ratio of the height of the tree to the number of nodes is logarithmic.
		Notice that when you are asked to determine whether a tree is height-balanced-n or balanced, if it is not you must demonstrate this fact by clicking any node which is out of balance.
		almost complete trees are the most balanced and that every almost complete tree is a balanced tree.
	www.cstc.org /data/resources/65/text5.html (273 words)

	Cell Tree
		The cell tree [16,17] is a height-balanced tree.
		The cell tree restricts the polyhedra to be partitions of a BSP (binary space partitioning), to avoid overlaps among sibling polyhedra.
		The splitting of a cell tree node is based on the plane sweep paradigm, which conducts plane sweeps across the node along l different directions to find a suitable splitting hyperplane.
	www.cs.umn.edu /research/shashi-group/paper_html/tkde/node22.html (312 words)

	f99mid.html
		Complete the BtNode method below which returns true for height-balanced trees, else false.
		The height of a tree is the length of a longest path from root to a leaf.
		Recall that the height of an empty tree is taken as -1.
	www.cs.rutgers.edu /~kaplan/503/exams/f99mid.html (323 words)

	OSE: Manual Pages / OTC_AVLTree
		This class is the root class of a height balanced AVL tree.
		The root class of a height balanced AVL tree.
		It is this class which holds the root node in the tree and maintains the population count for the tree.
	ose.sourceforge.net /browse-manual.php?manual=OTC_AVLTree&media=print (272 words)

	AVL tree
		An AVL tree is a balanced binary search tree where the height of the two child subtrees differ by at most one, otherwise known as height-balanced[?].
		Additions and deletions may require the tree to be rebalanced by one or more tree rotations.
		The AVL tree is named after its inventors, Adelson-Velskii[?] and Landis[?] (1962).
	www.eurofreehost.com /av/AVL_tree.html (142 words)

	Binary Search Trees and AVL Trees
		An AVL (height balanced) tree is shown beneath the ordinary binary search tree in the HTML FORM above.
		Balance of AVL (sub-)tree(s) is indicated by `' (balanced), `^' (right taller) and `v' (left taller).
		The roots of the trees are at the left-hand-side of the large TEXTAREA.
	www.csse.monash.edu /~lloyd/tildeAlgDS/Tree/Search (190 words)

	SICOMP Volume 8 Issue 1
		We determine the proportion of balanced nodes in this section of the tree, and find the average number of single and double rotations which occur at the minimum height during insertions.
		If $\bar B_n $ denotes the average number of balanced nodes in a random height-balanced tree with $n$ internal nodes, we show that $\frac{{10}}{{21}} (n + 1) \leqq \bar B_n \leqq \frac{6}{7}(n + 1) - 1$ for $n \geqq 6$.
		The collection of nodes nearest to the external nodes in a random height-balanced tree is analyzed.
	locus.siam.org /SICOMP/volume-08/art_0208003.html (130 words)

	R_Tree
		An R-tree is a height-balanced tree similar to a B-tree with index records in its leaf nodes containing pointers to data objects.
		For each such entry invoke FindLeaf on the tree whose root is pointed to by that entry until e is found or all entries have been checked.
		Function: Recursively descends tree from the root, collecting all nodes and entries contained in the tree.
	foto.hut.fi /research/TargetJr/Manpages/GeneralUtility/Basics/rtree.html (1329 words)

	CTTree - A template class to handle height-balanced binary trees (AVL trees)
		Height-balanced trees have the property that any two subtrees of a node differ in height by 1 at the maximum.
		Thus, the total height of a tree is kept minimal.
		Although this is rather uncommon in AVL tree applications, there are cases in which you may wish to change the organization of the tree to another key.
	pc1-archbo.bot.unibas.ch /~pascal/ttreemanual.html (3960 words)

	AVL tree
		Definition: A balanced binary search tree where the height of the two subtrees (children) of a node differs by at most one.
		See also left rotation, right rotation, red-black tree, height-balanced tree, B-tree, threaded tree, Fibonacci tree.
		Note: If necessary, the tree is rebalanced after insertions or deletions using rotations.
	www.darkridge.com /~jpr5/archive/dads/HTML/avltree.html (145 words)

	An English-Persian Dictionary of Algorithms and Data Structures
		See also AVL tree, red-black tree, height-balanced tree, weight-balanced tree, left rotation, right rotation, B-tree.
		After inserting or deleting a node, the tree may rebalanced with "rotations."
		Definition: A binary tree where no leaf is more than a certain amount farther from the root than any other.
	ce.sharif.edu /~dic-ads/d.php?r=balanced+binary+tree.1.14 (81 words)

	CSCI 3510 Practice questions for midterm exam 2
		If you were to use the standard (unbalanced) binary search tree insertion algorithm, show what this tree would look like after inserting 40.
		The binary search tree implementation that was discussed in class was nonpersistent; the insert operation, for example, changed the tree.
		It is possible to implement binary search trees in a persistent way also, so that they compute new trees from old ones.
	www.cs.ecu.edu /~karl/3510/sum00/exam2.html (613 words)

	f97final.html
		In the height-balanced tree shown below, lower-case letters represent nodes, and the upper-case letters represent subtrees.
		Why might you prefer - if you do - a binary heap implementation to using a height-balanced binary search tree to implement a heap?
		The initial subtree heights are A, C, D: h-1; B, F, G, H: h; E: h+1.
	www.cs.rutgers.edu /~kaplan/503/exams/f97final.html (337 words)

	Data Structures Review Questions, Sections 7.1 and 7.2
		Write a member function that searches a binary search tree for a given key, inserts the key if it is not already in the tree, and always returns a pointer to the node containing the item.
		Rebalance the tree, labeling each node or subtree symbol (triangle) with the correct symbol, and labeling each edge with the correct weight.
		Describe the balancing method for obtaining a near-optimal search tree without going into details.
	www.csm.astate.edu /~rossa/cs3363/rvw7a.html (297 words)

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