477:
4553:
1462:. If the tree is empty, then the node is inserted as the root of the tree. If the tree is not empty, then we go down the root, and recursively go down the tree searching for the location to insert the new node. This traversal is guided by the comparison function. In this case, the node always replaces a NULL reference (left or right) of an external node in the tree i.e., the node is either made a left-child or a right-child of the external node.
5002:
2399:
subtree is unbalanced and needs to be rotated. (Unlike insertion where a rotation always balances the tree, after delete, there may be BF(Z) ā 0 (see figures 2 and 3), so that after the appropriate single or double rotation the height of the rebalanced subtree decreases by one meaning that the tree has to be rebalanced again on the next higher level.) The various cases of rotations are described in section
33:
7135:
3539:. The increase in height can increase the height of its ancestors, possibly invalidating the AVL invariant of those nodes. This can be fixed either with a double rotation if invalid at the parent or a single left rotation if invalid higher in the tree, in both cases restoring the height for any further ancestor nodes.
3311:
If the balance factor temporarily becomes Ā±2, this has to be repaired by an appropriate rotation. It depends on the balance factor of the sibling Z (the higher child tree in figure 2) whether the height of the subtree decreases by one āand the retracing needs to continueā or does not change (if Z has
1477:
For each node checked, if the temporary balance factor remains in the range from ā1 to +1 then only an update of the balance factor and no rotation is necessary. However, if the temporary balance factor is Ā±2, the subtree rooted at this node is AVL unbalanced, and a rotation is needed. With insertion
5661:
Both AVL trees and redāblack (RB) trees are self-balancing binary search trees and they are related mathematically. Indeed, every AVL tree can be colored redāblack, but there are RB trees which are not AVL balanced. For maintaining the AVL (or RB) tree's invariants, rotations play an important role.
5693:
on average. RB trees require storing one bit of information (the color) in each node, while AVL trees mostly use two bits for the balance factor, although, when stored at the children, one bit with meaning Ā«lower than siblingĀ» suffices. The bigger difference between the two data structures is their
4225:
If during a modifying operation the height difference between two child subtrees changes, this may, as long as it is < 2, be reflected by an adaption of the balance information at the parent. During insert and delete operations a (temporary) height difference of 2 may arise, which means that the
2398:
Since with a single deletion the height of an AVL subtree cannot decrease by more than one, the temporary balance factor of a node will be in the range from ā2 to +2. If the balance factor remains in the range from ā1 to +1 it can be adjusted in accord with the AVL rules. If it becomes Ā±2 then the
469:
2339:
In order to update the balance factors of all nodes, first observe that all nodes requiring correction lie from child to parent along the path of the inserted leaf. If the above procedure is applied to nodes along this path, starting from the leaf, then every node in the tree will again have a
1465:
After this insertion, if a tree becomes unbalanced, only ancestors of the newly inserted node are unbalanced. This is because only those nodes have their sub-trees altered. So it is necessary to check each of the node's ancestors for consistency with the invariants of AVL trees: this is called
4997:
As the figure shows, before an insertion, the leaf layer was at level h+1, temporarily at level h+2 and after the double rotation again at level h+1. In case of a deletion, the leaf layer was at level h+2 and after the double rotation it is at level h+1, so that the height of the rotated tree
1431:
links (particularly when navigating from the rightmost leaf of the root's left subtree to the root or from the root to the leftmost leaf of the root's right subtree; in the AVL tree of figure 1, navigating from node P to the next-to-the-right node Q takes 3 steps). Since there are
779:
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The result of the first, the right, rotation is shown in the middle third of the figure. (With respect to the balance factors, this rotation is not of the same kind as the other AVL single rotations, because the height difference between Y and
610:
time for the basic operations. For lookup-intensive applications, AVL trees are faster than redāblack trees because they are more strictly balanced. Similar to redāblack trees, AVL trees are height-balanced. Both are, in general, neither
6543:
906:
Balance factors can be kept up-to-date by knowing the previous balance factors and the change in height ā it is not necessary to know the absolute height. For holding the AVL balance information, two bits per node are sufficient.
4540:
As the figure shows, before an insertion, the leaf layer was at level h+1, temporarily at level h+2 and after the rotation again at level h+1. In case of a deletion, the leaf layer was at level h+2, where it is again, when
1181:
4233:
Let X be the node that has a (temporary) balance factor of ā2 or +2. Its left or right subtree was modified. Let Z be the child with the higher subtree (see figures 2 and 3). Note that both children are in AVL shape by
6151:
6079:
4083:
5883:
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However, the balance information can be kept in the child nodes as one bit indicating whether the parent is higher by 1 or by 2; thereby higher by 2 cannot occur for both children. This way the AVL tree is a
5946:
6016:
1039:
7139:
3775:, all the subtrees on the left side are merged bottom-up using keys on the path as intermediate nodes from bottom to top to form the left tree, and the right part is asymmetric. The cost of
2392:. There, the effective deletion of the subject node or the replacement node decreases the height of the corresponding child tree either from 1 to 0 or from 2 to 1, if that node had a child.
6413:
2349:
If the balance factor temporarily becomes Ā±2, this has to be repaired by an appropriate rotation after which the subtree has the same height as before (and its root the balance factor 0).
1288:
1405:
nodes of the tree visits each link exactly twice: one downward visit to enter the subtree rooted by that node, another visit upward to leave that node's subtree after having explored it.
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but without the middle key. Based on the new functions for union, intersection or difference, either one key or multiple keys can be inserted to or deleted from the AVL tree. Since
859:
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608:
457:
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subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Lookup, insertion, and deletion all take
4994:
is only 1.) The result of the final left rotation is shown in the lower third of the figure. Five links (thick edges in figure 3) and three balance factors are to be updated.
4241:
In case of insertion this insertion has happened to one of Z's children in a way that Z's height has increased. In case of deletion this deletion has happened to the sibling t
896:
1338:
4230:, because they move the keys only "vertically", so that the ("horizontal") in-order sequence of the keys is fully preserved (which is essential for a binary-search tree).
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Since with a single insertion the height of an AVL subtree cannot increase by more than one, the temporary balance factor of a node after an insertion will be in the range
178:
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6642:
6565:
6372:
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relation AVL/RB ā0.720 of the maximal heights. For insertions and deletions, Ben Pfaff shows in 79 measurements a relation of AVL/RB between 0.677 and 1.077 with
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4103:
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The result of the left rotation is shown in the lower half of the figure. Three links (thick edges in figure 2) and two balance factors are to be updated.
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Starting at this subtree, it is necessary to check each of the ancestors for consistency with the invariants of AVL trees. This is called "retracing".
1104:
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6091:
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562:, who published it in their 1962 paper "An algorithm for the organization of information". It is the oldest self-balancing binary search tree
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inspections and/or updates to AVL balance factors (or RB colors). RB insertions and deletions and AVL insertions require from zero to three
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If the balance factor becomes 0 (it must have been Ā±1) then the height of the subtree decreases by one and the retracing needs to continue.
8032:
7150:
3367:
operations on insertions or deletions can be implemented based on these set functions. These set operations rely on two helper operations,
6261:
3305:
The retracing can stop if the balance factor becomes Ā±1 (it must have been 0) meaning that the height of that subtree remains unchanged.
543:
is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more
5835:
1047:
4138:. More importantly, since the recursive calls to union, intersection or difference are independent of each other, they can be executed
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3543:
will therefore require at most two rotations. The cost of this function is the difference of the heights between the two input trees.
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5951:
956:
472:
Animation showing the insertion of several elements into an AVL tree. It includes left, right, left-right and right-left rotations.
3351:
In addition to the single-element insert, delete and lookup operations, several set operations have been defined on AVL trees:
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3983:
is presumed to return two trees: one holding the keys less its input key, one holding the greater keys. (The algorithm is
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As a read-only operation the traversal of an AVL tree functions the same way as on any other binary tree. Exploring all
8086:
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If the balance factor becomes Ā±1 then the height of the subtree increases by one and the retracing needs to continue.
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were of same height. Otherwise the leaf layer reaches level h+1, so that the height of the rotated tree decreases.
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of Z (i.e., left child when Z is right child, or right child when Z is left child) is not higher than its sibling t
2343:
The retracing can stop if the balance factor becomes 0 implying that the height of that subtree remains unchanged.
1347:
Read-only operations of an AVL tree involve carrying out the same actions as would be carried out on an unbalanced
1245:
4011:
but does not deal with the balancing criteria of AVL trees directly, such an implementation is usually called the
1485:
In figure 1, by inserting the new node Z as a child of node X the height of that subtree Z increases from 0 to 1.
774:{\displaystyle {\text{BF}}(X):={\text{Height}}({\text{RightSubtree}}(X))-{\text{Height}}({\text{LeftSubtree}}(X))}
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7059:
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Figure 3 shows a Right Left situation. In its upper third, node X has two child trees with a balance factor of
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Figure 2 shows a Right Right situation. In its upper half, node X has two child trees with a balance factor of
4148:
638:
7701:
4249:'s height being already lower has decreased. (This is the only case where Z's balance factor may also be 0.)
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3375:. With the new operations, the implementation of AVL trees can be more efficient and highly-parallelizable.
5819:{\displaystyle {\begin{array}{ll}h&\leqq \;c\log _{2}(n+d)+b\\&<\;c\log _{2}(n+2)+b\end{array}}}
1359:
Searching for a specific key in an AVL tree can be done the same way as that of any balanced or unbalanced
205:
7815:
7798:
4217:, the join-based implementation has the same computational DAG as single-element insertion and deletion.
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790:
1371:) on the set of keys. The number of comparisons required for successful search is limited by the height
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is created to replace c. The new node satisfies the AVL invariant, and its height is one greater than
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1363:. In order for search to work effectively it has to employ a comparison function which establishes a
43:
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Adelson-Velsky, Georgy; Landis, Evgenii (1962). "An algorithm for the organization of information".
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152:
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When inserting a node into an AVL tree, you initially follow the same process as inserting into a
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92:
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7177:
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Blelloch, Guy E.; Ferizovic, Daniel; Sun, Yihan (2016), "Just join for parallel ordered sets",
4235:
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7908:
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4974:(with the consequence that they are of different height) or by a height decrease of subtree t
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115:
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4966:. This can happen by the insertion of Y itself or a height increase of one of its subtrees t
7893:
7334:
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6538:{\displaystyle {\tfrac {1}{2}}-\mu \leq {\tfrac {|N_{l}|}{|N|+1}}\leq {\tfrac {1}{2}}+\mu }
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4139:
1417:
612:
6550:
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618:
8:
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6731:. New Delhi, India: University Science Press, an imprint of Laxmi Publications Pvt. Ltd.
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constant time. Some instances of exploring these "nearby" nodes require traversing up to
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The height of the subtree rooted by N has decreased by 1. It is already in AVL shape.
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The height of the subtree rooted by Z has increased by 1. It is already in AVL shape.
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In the worst case, even without rotations, AVL or RB insertions or deletions require
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6688:(2. ed., 6. printing, newly updated and rev. ed.). Boston : Addison-Wesley.
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parent subtree has to be "rebalanced". The given repair tools are the so-called
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1176:{\displaystyle b:={\frac {\log _{2}5}{2\log _{2}\varphi }}-2\approx \;-0.3277.}
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on average) on the way back to the root, so the operation can be completed in
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on average) on the way back to the root, so the operation can be completed in
2329:// Unless loop is left via break, the height of the total tree increases by 1.
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AVL trees are not weight-balanced? (meaning: AVL trees are not Ī¼-balanced?)
6146:{\displaystyle {\begin{array}{ll}h&\leqq \;2\log _{2}(n+1)\end{array}}}
5886:
1098:
670:; that is, sibling nodes can have hugely differing numbers of descendants.
551:
7017:
4962:. But unlike figure 2, the inner child Y of Z is higher than its sibling t
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The algorithm for intersection or difference is similar, but requires the
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6074:{\displaystyle d:=1+{\tfrac {1}{\varphi ^{4}{\sqrt {5}}}}\approx 1.065}
4078:{\displaystyle {\text{O}}\left(m\log \left({n \over m}+1\right)\right)}
476:
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as the code below shows, the adequate rotation immediately perfectly
7103:
6802:
Schaum's outline of theory and problems of data structures with Java
4552:
3751:
To split an AVL tree into two smaller trees, those smaller than key
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will be found on the left of the path, and all values greater than
2388:
The preliminary steps for deleting a node are described in section
468:
7686:
5878:{\displaystyle \varphi :={\tfrac {1+{\sqrt {5}}}{2}}\approx 1.618}
5001:
1090:{\displaystyle \varphi :={\tfrac {1+{\sqrt {5}}}{2}}\approx 1.618}
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time, thus equally constant on average. AVL deletions requiring
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6758:
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5941:{\displaystyle c:={\tfrac {1}{\log _{2}\varphi }}\approx 1.440,}
4018:
The complexity of each of union, intersection and difference is
3295:// If (b != 0) the height of the total tree decreases by 1.
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6011:{\displaystyle b:={\tfrac {c}{2}}\log _{2}5-2\approx \;-0.328,}
3809:(T = nil) return (nil, false, nil) (L,m,R) = expose(T)
1034:{\displaystyle \log _{2}(n+1)\leq h<\log _{\varphi }(n+2)+b}
6720:
4498:
The cost of a rotation, either simple or double, is constant.
930:(counted as the maximal number of levels) of an AVL tree with
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7279:
6859:(3rd ed.). Boston: Pearson Addison-Wesley. p. 145.
6753:
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6159:
AVL trees are more rigidly balanced than RB trees with an
3312:
the balance factor 0) and the whole tree is in AVL-shape.
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3763:
into the AVL. After this insertion, all values less than
573:
because both support the same set of operations and take
3987:, but an in-place destructive version exists as well.)
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4526:. In the latter case, also the pale situation where t
4197:
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3346:
2830:// (N == right_child(X)): The right subtree decreases
2176:// Height does not change: Height(N) == old Height(X)
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4518:. This can happen by a height increase of subtree t
4252:There are four possible variants of the violation:
3731:. Node(l,k,r) means to create a node of left child
691:of a node X is defined to be the height difference
7006:Dinesh P. Mehta; Sartaj Sahni, eds. (2017-12-15).
6977:. Berlin, Heidelberg: Springer Berlin Heidelberg.
6893:Symposium on Parallel Algorithms and Architectures
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2323:// There is no fall thru, only break; or continue;
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7075:. Pages 458ā475 of section 6.2.3: Balanced Trees.
6331:"Performance Analysis of BSTs in System Software"
5656:
3887:Pseudocode implementation for the Union algorithm
3795:Pseudocode implementation for the Split algorithm
3707:Here Height(v) is the height of a subtree (node)
3492:for more than one (the other case is symmetric).
3448:. If the two trees differ by height at most one,
1891:// Z == left_child(X): the left subtree increases
1466:"retracing". This is achieved by considering the
8058:
3825:(k>m) (L',b,R') = Split(R, k)
3585:) if (Height(T') <= Height(l)+1) then
3551:Pseudocode implementation for the Join algorithm
3289:// Height(N) decreases by 1 (== old Height(X)-1)
523:time in both the average and worst cases, where
6970:
4888:// 2nd case happens with insertion or deletion:
4789:// only happens with deletion, not insertion:
3817:(k<m) (L',b,R') = Split(L,k)
3759:, first draw a path from the root by inserting
1408:Once a node has been found in an AVL tree, the
7012:(2 ed.). New York: Chapman and Hall/CRC.
6952:National Institute of Standards and Technology
6729:Mastering data structures through 'C' language
6408:{\displaystyle 0\leq \mu \leq {\tfrac {1}{2}}}
4978:. In the latter case, it may also occur that t
4323:And the rebalancing is performed differently:
3404:will return a tree containing all elements in
2407:Invariant of the retracing loop for a deletion
784:of its two child sub-trees rooted by node X.
7702:
7178:
6941:
6886:
6884:
6857:Data structures and algorithm analysis in C++
3593:rotateLeft(Node(l, k', rotateRight(T')))
1377:and for unsuccessful search is very close to
1283:{\displaystyle \{F_{n}\}_{n\in \mathbb {N} }}
480:Fig. 1: AVL tree with balance factors (green)
7151:Dictionary of Algorithms and Data Structures
7009:Handbook of Data Structures and Applications
6948:Dictionary of Algorithms and Data Structures
5040:Code snippet of a right-left double rotation
1263:
1249:
6597:is the number of nodes below the tree with
6262:Proceedings of the USSR Academy of Sciences
3518:. At this point a new node with left child
3452:simply create a new node with left subtree
3121:// N is the new root of the rotated subtree
2173:// N is the new root of the rotated subtree
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6769:. Cambridge: Cambridge University Press.
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3286:// Height does not change: Leave the loop
2956:// Double rotation: Left(Z) then Right(X)
2659:// Double rotation: Right(Z) then Left(X)
2008:// Double rotation: Left(Z) then Right(X)
1720:// Double rotation: Right(Z) then Left(X)
1439:links in any tree, the amortized cost is
1274:
508:. In an AVL tree, the heights of the two
69:Learn how and when to remove this message
7192:
6835:
6833:
6831:
6324:
6322:
5000:
4551:
4184:{\displaystyle {\text{O}}(\log m\log n)}
3771:will be found on the right. By applying
3601:) T'' = Node(l, k', T')
3055:// Nās height decrease is absorbed at X.
2761:// Nās height decrease is absorbed at X.
2107:// Zās height increase is absorbed at X.
1819:// Zās height increase is absorbed at X.
663:{\displaystyle \mu \leq {\tfrac {1}{2}}}
475:
467:
7067:, Third Edition. Addison-Wesley, 1997.
6971:Mehlhorn, Kurt; Sanders, Peter (2008).
6797:
6676:
6674:
6672:
6670:
4591:Z = right child of X, Z is right-heavy
4584:X = root of subtree to be rotated left
4418:Thereby, the situations are denoted as
4317:child of its parent X and BF(Z) > 0
4301:child of its parent X and BF(Z) < 0
898:is sometimes simply called "balanced".
14:
8059:
6999:
6222:
4574:Code snippet of a simple left rotation
3259:// N is the new root of the total tree
3118:// After a rotation adapt parent link:
2314:// N is the new root of the total tree
2170:// After a rotation adapt parent link:
1492:of the retracing loop for an insertion
1183:This is because an AVL tree of height
861:is called "right-heavy", and one with
7690:
7166:
7079:Haeupler, Bernhard; Sen, Siddhartha;
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6680:
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6319:
5689:rotations in the worst case are also
5648:// return new root of rotated subtree
4945:// return new root of rotated subtree
6667:
5126:// Z is by 2 higher than its sibling
4822:// t23 has been of same height as t4
4660:// Z is by 2 higher than its sibling
4522:or by a height decrease of subtree t
4285:child of its parent X and BF(Z) ā¤ 0
4269:child of its parent X and BF(Z) ā„ 0
3630:) /* symmetric to JoinRightAVL */
2500:// Save parent of X around rotations
1951:// Save parent of X around rotations
1663:// Save parent of X around rotations
1504:Example code for an insert operation
550:The AVL tree is named after its two
26:
7716:
3999:helper routine that is the same as
3787:, order of the height of the tree.
2992:// After rotation adapt parent link
2866:// ==> the temporary BF(X) == -2
2695:// After rotation adapt parent link
2569:// ==> the temporary BF(X) == +2
2419:Example code for a delete operation
2044:// After rotation adapt parent link
1927:// ==> the temporary BF(X) == -2
1756:// After rotation adapt parent link
1639:// ==> the temporary BF(X) == +2
1467:
854:{\displaystyle {\text{BF}}(X)>0}
817:{\displaystyle {\text{BF}}(X)<0}
24:
7048:
5050:X = root of subtree to be rotated
4953:
4501:
3347:Set operations and bulk operations
2869:// ==> rebalancing is required.
2665:// Right Right Case (see figure 2)
2632:// Right Left Case (see figure 3)
2572:// ==> rebalancing is required.
2503:// BF(X) has not yet been updated!
1930:// ==> rebalancing is required.
1726:// Right Right Case (see figure 2)
1693:// Right Left Case (see figure 3)
1642:// ==> rebalancing is required.
603:{\displaystyle {\text{O}}(\log n)}
569:AVL trees are often compared with
452:{\displaystyle {\text{O}}(\log n)}
414:{\displaystyle {\text{O}}(\log n)}
369:{\displaystyle {\text{O}}(\log n)}
331:{\displaystyle {\text{O}}(\log n)}
286:{\displaystyle {\text{O}}(\log n)}
248:{\displaystyle {\text{O}}(\log n)}
25:
8098:
7127:
6617:as root (including the root) and
6354:Thereby: A Binary Tree is called
4486:is repaired by a double rotation
4465:is repaired by a simple rotation
2479:// Loop (possibly up to the root)
1570:// Loop (possibly up to the root)
824:is called "left-heavy", one with
678:
506:self-balancing binary search tree
81:Self-balancing binary search tree
7138: This article incorporates
7133:
6844:. Free Software Foundation, Inc.
5707:an AVL tree's height is at most
5150:// Y is by 1 higher than sibling
5057:Z = its right child, left-heavy
3813:(k = m) return (L, true, R)
2400:
1479:
891:{\displaystyle {\text{BF}}(X)=0}
31:
7060:The Art of Computer Programming
7034:
6964:
6935:
6848:
6567:is minimal with this property.
5079:new root of rebalanced subtree
4613:new root of rebalanced subtree
3711:. (l,k,r) = expose(v) extracts
3605:(Height(T') <= Height(l)+1)
3430:to be greater than all keys in
2340:balance factor of ā1, 0, or 1.
7092:ACM Transactions on Algorithms
7041:Redāblack tree#Proof of bounds
6974:Algorithms and Data Structures
6702:
6583:
6575:
6500:
6492:
6485:
6470:
6344:
6281:
6252:
6136:
6124:
6086:a RB tree's height is at most
5803:
5791:
5757:
5745:
5657:Comparison to other structures
4455:The balance violation of case
4433:(= balance) come from the set
4220:
4178:
4157:
4125:
4116:
3323:for lookup, plus a maximum of
2360:for lookup, plus a maximum of
1603:// The right subtree increases
1333:{\displaystyle F_{1}=F_{2}=1.}
1022:
1010:
985:
973:
879:
873:
842:
836:
805:
799:
768:
765:
759:
751:
740:
737:
731:
723:
712:
706:
597:
585:
446:
434:
408:
396:
363:
351:
325:
313:
280:
268:
242:
230:
167:
161:
40:This article needs editing to
13:
1:
6714:
6216:
5064: with height ==
4598: with height ==
4510:. Moreover, the inner child t
3755:, and those greater than key
3727:'s root, and the right child
3581:)+1) T' = Node(c, k, T
3439:and smaller than all keys in
2890:// Sibling of N (higher by 2)
2593:// Sibling of N (higher by 2)
2533:// the left subtree decreases
1342:
901:
673:
173:{\displaystyle {\text{O}}(n)}
6942:Paul E. Black (2015-04-13).
6276:Soviet Mathematics - Doklady
3821:(L', b, Join(R', m, R))
3589:Node(l, k', T') else
1394:
1354:
950:nodes lies in the interval:
7:
8033:Directed acyclic word graph
7799:Double-ended priority queue
6798:Hubbard, John Rast (2000).
6174:
4412:(mirror-image of figure 3)
4368:(mirror-image of figure 2)
4131:{\displaystyle n\;(\geq m)}
4013:"join-based" implementation
3837:The union of two AVL trees
3569:) (l, k', c) = expose(T
3496:follows the right spine of
3103:// Height(N) decreases by 1
2989:// Single rotation Right(X)
2809:// Height(N) decreases by 1
2390:Binary search tree#Deletion
2155:// Height(Z) increases by 1
2041:// Single rotation Right(X)
1870:// Height(Z) increases by 1
1573:// BF(X) has to be updated:
10:
8103:
6855:Weiss, Mark Allen (2006).
6644:is the left child node of
6301:. Addison-Wesley. p.
3474:. Otherwise, suppose that
2692:// Single rotation Left(X)
1753:// Single rotation Left(X)
8087:Amortized data structures
8041:
8013:
7907:
7869:
7826:
7745:
7724:
7588:
7467:
7414:
7343:
7200:
6983:10.1007/978-3-540-77978-0
6895:, ACM, pp. 253ā264,
6806:. New York: McGraw-Hill.
3829:(Join(L, m, L'), b, R'))
3597:T' = JoinRightAVL(c, k, T
3577:(Height(c) <= Height(T
2383:
1453:
1235:{\displaystyle F_{h+2}-1}
379:
296:
213:
192:
185:
144:
137:
133:
125:
111:
103:
91:
86:
7765:Retrieval Data Structure
7641:Left-child right-sibling
6767:Advanced data structures
6715:Haeupler, Sen and Tarjan
6329:Pfaff, Ben (June 2004).
5084:
5005:Fig. 3: Double rotation
4986:are of the same height.
4786:// 1st case, BF(Z) == 0,
4618:
4530:has the same height as t
4429:(= child direction) and
4400:ā¹ X is rebalanced with a
4378:ā¹ X is rebalanced with a
4356:ā¹ X is rebalanced with a
4334:ā¹ X is rebalanced with a
2425:
1510:
1416:node can be accessed in
42:comply with Knowledge's
8046:List of data structures
8023:Binary decision diagram
7471:data partitioning trees
7429:C-trie (compressed ADT)
6911:10.1145/2935764.2935768
6211:List of data structures
5393:// 1st case, BF(Y) == 0
4556:Fig. 2: Simple rotation
4245:of Z in a way so that t
4085:for AVL trees of sizes
3509:which is balanced with
492:(named after inventors
8028:Directed acyclic graph
7140:public domain material
6658:
6638:
6611:
6591:
6561:
6539:
6429:
6409:
6368:
6147:
6075:
6012:
5942:
5879:
5820:
5036:
4876:// t4 now lower than X
4570:
4211:
4185:
4132:
4099:
4079:
1450:, or approximately 2.
1334:
1284:
1236:
1197:
1177:
1091:
1035:
944:
924:
892:
855:
818:
775:
664:
629:
604:
537:
481:
473:
453:
415:
370:
332:
287:
249:
174:
7085:"Rank-balanced trees"
7065:Sorting and Searching
7018:10.1201/9781315119335
6765:Brass, Peter (2008).
6727:Dixit, J. B. (2010).
6686:Sorting and searching
6659:
6639:
6637:{\displaystyle N_{l}}
6612:
6592:
6562:
6540:
6430:
6410:
6369:
6274:by Myron J. Ricci in
6148:
6076:
6013:
5943:
5880:
5821:
5674:rotations and run in
5004:
4555:
4212:
4186:
4133:
4100:
4080:
3315:The time required is
2352:The time required is
1335:
1294:with the seed values
1285:
1237:
1198:
1178:
1092:
1036:
945:
925:
893:
856:
819:
776:
665:
630:
605:
556:Georgy Adelson-Velsky
538:
479:
471:
454:
416:
371:
333:
288:
250:
175:
116:Georgy Adelson-Velsky
7894:Unrolled linked list
7651:Log-structured merge
7194:Tree data structures
6711:"rank balanced" tree
6648:
6621:
6601:
6571:
6560:{\displaystyle \mu }
6551:
6441:
6419:
6415:, if for every node
6378:
6367:{\displaystyle \mu }
6358:
6278:, 3:1259ā1263, 1962.
6186:Weight-balanced tree
6092:
6022:
5952:
5893:
5836:
5713:
4236:induction hypothesis
4195:
4149:
4109:
4089:
4022:
3968:.root, Union(right(t
1298:
1246:
1207:
1187:
1105:
1048:
957:
934:
914:
865:
828:
791:
698:
639:
628:{\displaystyle \mu }
619:
577:
527:
426:
388:
343:
305:
260:
222:
153:
7942:Self-balancing tree
6840:Pfaff, Ben (2004).
6590:{\displaystyle |N|}
6338:Stanford University
6272:English translation
5697:For a tree of size
5147:// Inner child of Z
5066:Height(LeftSubtree(
4681:// Inner child of Z
4600:Height(LeftSubtree(
4210:{\displaystyle m=1}
2566:// X is right-heavy
1636:// X is right-heavy
51:improve the content
7922:Binary search tree
7787:Double-ended queue
7616:Fractal tree index
7211:associative arrays
7098:(4): Art. 30, 26,
6654:
6634:
6607:
6587:
6557:
6535:
6527:
6512:
6454:
6425:
6405:
6403:
6364:
6143:
6141:
6071:
6063:
6008:
5971:
5938:
5927:
5875:
5867:
5816:
5814:
5037:
4571:
4207:
4181:
4128:
4095:
4075:
3855:representing sets
3739:, and right child
3465:and right subtree
3331:retracing levels (
2929:// Left Right Case
2863:// X is left-heavy
2368:retracing levels (
1981:// Left Right Case
1924:// X is left-heavy
1460:Binary Search Tree
1361:binary search tree
1349:binary search tree
1330:
1292:Fibonacci sequence
1280:
1232:
1203:contains at least
1193:
1173:
1087:
1079:
1031:
940:
920:
888:
851:
814:
771:
660:
658:
635:-balanced for any
625:
600:
533:
482:
474:
449:
411:
366:
328:
283:
245:
170:
8077:Soviet inventions
8067:1962 in computing
8054:
8053:
7856:Hashed array tree
7755:Associative array
7684:
7683:
7081:Tarjan, Robert E.
7027:978-1-315-11933-5
6992:978-3-540-77977-3
6920:978-1-4503-4210-0
6657:{\displaystyle N}
6610:{\displaystyle N}
6526:
6511:
6453:
6435:, the inequality
6428:{\displaystyle N}
6402:
6289:Sedgewick, Robert
6248:on July 31, 2019.
6062:
6059:
5970:
5926:
5866:
5860:
5083:
5082:
4849:// t23 now higher
4617:
4616:
4476:whereas the case
4416:
4415:
4321:
4320:
4155:
4098:{\displaystyle m}
4057:
4028:
3992:
3991:
3956:Join(Union(left(t
3834:
3833:
3748:
3747:
3382:on two AVL trees
3302:
3301:
3064:// Leave the loop
2962:// Left Left Case
2770:// Leave the loop
2336:
2335:
2116:// Leave the loop
2014:// Left Left Case
1828:// Leave the loop
1383:, so both are in
1196:{\displaystyle h}
1155:
1078:
1072:
943:{\displaystyle n}
923:{\displaystyle h}
871:
834:
797:
757:
749:
729:
721:
704:
657:
583:
566:to be invented.
536:{\displaystyle n}
466:
465:
462:
461:
432:
394:
349:
311:
266:
228:
159:
79:
78:
71:
16:(Redirected from
8094:
7879:Association list
7711:
7704:
7697:
7688:
7687:
7187:
7180:
7173:
7164:
7163:
7159:
7137:
7136:
7122:
7089:
7043:
7038:
7032:
7031:
7003:
6997:
6996:
6968:
6962:
6961:
6959:
6958:
6939:
6933:
6931:
6904:
6888:
6879:
6878:
6852:
6846:
6845:
6837:
6826:
6825:
6805:
6795:
6789:
6788:
6762:
6751:
6750:
6724:
6718:
6706:
6700:
6699:
6682:Knuth, Donald E.
6678:
6665:
6663:
6661:
6660:
6655:
6643:
6641:
6640:
6635:
6633:
6632:
6616:
6614:
6613:
6608:
6596:
6594:
6593:
6588:
6586:
6578:
6566:
6564:
6563:
6558:
6544:
6542:
6541:
6536:
6528:
6519:
6513:
6510:
6503:
6495:
6489:
6488:
6483:
6482:
6473:
6467:
6455:
6446:
6434:
6432:
6431:
6426:
6414:
6412:
6411:
6406:
6404:
6395:
6374:-balanced, with
6373:
6371:
6370:
6365:
6348:
6342:
6341:
6335:
6326:
6317:
6316:
6296:
6294:"Balanced Trees"
6285:
6279:
6270:
6256:
6250:
6249:
6244:. Archived from
6240:Eric Alexander.
6237:
6152:
6150:
6149:
6144:
6142:
6120:
6119:
6080:
6078:
6077:
6072:
6064:
6061:
6060:
6055:
6053:
6052:
6039:
6017:
6015:
6014:
6009:
5982:
5981:
5972:
5963:
5947:
5945:
5944:
5939:
5928:
5925:
5918:
5917:
5904:
5884:
5882:
5881:
5876:
5868:
5862:
5861:
5856:
5847:
5825:
5823:
5822:
5817:
5815:
5787:
5786:
5769:
5741:
5740:
5703:
5692:
5688:
5680:
5669:
5652:
5649:
5646:
5643:
5640:
5637:
5634:
5631:
5628:
5625:
5622:
5619:
5616:
5613:
5612:// t4 now higher
5610:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5586:
5583:
5580:
5577:
5574:
5571:
5568:
5565:
5564:// t2 was higher
5562:
5559:
5556:
5553:
5550:
5547:
5544:
5541:
5538:
5535:
5532:
5531:// t1 now higher
5529:
5526:
5523:
5520:
5517:
5514:
5511:
5508:
5505:
5504:// t3 was higher
5502:
5499:
5496:
5493:
5490:
5487:
5484:
5481:
5478:
5475:
5472:
5469:
5466:
5463:
5460:
5457:
5454:
5451:
5448:
5445:
5442:
5439:
5436:
5433:
5430:
5427:
5424:
5421:
5418:
5415:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5292:
5289:
5286:
5283:
5280:
5277:
5274:
5271:
5268:
5265:
5262:
5259:
5256:
5253:
5250:
5247:
5244:
5241:
5238:
5235:
5232:
5229:
5226:
5223:
5220:
5217:
5214:
5211:
5208:
5205:
5202:
5199:
5196:
5193:
5190:
5187:
5184:
5181:
5178:
5175:
5172:
5169:
5166:
5163:
5160:
5157:
5154:
5151:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5112:
5109:
5106:
5103:
5100:
5097:
5094:
5093:rotate_RightLeft
5091:
5088:
5071:
5067:
5044:
5043:
5007:rotate_RightLeft
4961:
4949:
4946:
4943:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4913:
4910:
4907:
4904:
4901:
4898:
4895:
4892:
4889:
4886:
4883:
4880:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4820:
4817:
4814:
4811:
4808:
4805:
4802:
4799:
4796:
4793:
4790:
4787:
4784:
4781:
4778:
4775:
4772:
4769:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4712:
4709:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4664:
4661:
4658:
4655:
4652:
4649:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4605:
4601:
4578:
4577:
4509:
4495:
4490:
4485:
4475:
4469:
4464:
4454:
4443:
4424:
4409:
4408:rotate_LeftRight
4387:
4386:rotate_RightLeft
4365:
4343:
4326:
4325:
4255:
4254:
4216:
4214:
4213:
4208:
4190:
4188:
4187:
4182:
4156:
4153:
4137:
4135:
4134:
4129:
4104:
4102:
4101:
4096:
4084:
4082:
4081:
4076:
4074:
4070:
4069:
4065:
4058:
4050:
4029:
4026:
3883:
3882:
3878:
3869:that represents
3868:
3862:
3858:
3854:
3845:
3805:Split(T, k)
3791:
3790:
3786:
3770:
3766:
3762:
3758:
3754:
3742:
3738:
3734:
3730:
3726:
3722:
3718:
3714:
3710:
3616:rotateLeft(T'')
3547:
3546:
3538:
3534:
3526:and right child
3525:
3521:
3517:
3508:
3504:
3491:
3482:
3473:
3464:
3460:
3447:
3438:
3429:
3425:
3421:
3412:
3403:
3399:
3390:
3342:
3334:
3330:
3322:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2938:rotate_LeftRight
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
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2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
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2810:
2807:
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2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2641:rotate_RightLeft
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
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2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2415:
2414:
2379:
2371:
2367:
2359:
2330:
2327:
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2264:
2261:
2258:
2255:
2252:
2249:
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2240:
2237:
2234:
2231:
2228:
2225:
2222:
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2216:
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2201:
2198:
2195:
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2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
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2102:
2099:
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2087:
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2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1990:rotate_LeftRight
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
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1787:
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1781:
1778:
1775:
1772:
1769:
1766:
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1760:
1757:
1754:
1751:
1748:
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1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1702:rotate_RightLeft
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1500:
1499:
1476:
1449:
1438:
1430:
1404:
1390:
1382:
1376:
1339:
1337:
1336:
1331:
1323:
1322:
1310:
1309:
1289:
1287:
1286:
1281:
1279:
1278:
1277:
1261:
1260:
1241:
1239:
1238:
1233:
1225:
1224:
1202:
1200:
1199:
1194:
1182:
1180:
1179:
1174:
1156:
1154:
1147:
1146:
1133:
1126:
1125:
1115:
1096:
1094:
1093:
1088:
1080:
1074:
1073:
1068:
1059:
1040:
1038:
1037:
1032:
1006:
1005:
969:
968:
949:
947:
946:
941:
929:
927:
926:
921:
897:
895:
894:
889:
872:
869:
860:
858:
857:
852:
835:
832:
823:
821:
820:
815:
798:
795:
780:
778:
777:
772:
758:
755:
750:
747:
730:
727:
722:
719:
705:
702:
669:
667:
666:
661:
659:
650:
634:
632:
631:
626:
609:
607:
606:
601:
584:
581:
542:
540:
539:
534:
522:
486:computer science
458:
456:
455:
450:
433:
430:
420:
418:
417:
412:
395:
392:
375:
373:
372:
367:
350:
347:
337:
335:
334:
329:
312:
309:
292:
290:
289:
284:
267:
264:
254:
252:
251:
246:
229:
226:
179:
177:
176:
171:
160:
157:
139:Space complexity
135:
134:
126:Complexities in
84:
83:
74:
67:
63:
60:
54:
35:
34:
27:
21:
8102:
8101:
8097:
8096:
8095:
8093:
8092:
8091:
8057:
8056:
8055:
8050:
8037:
8009:
7903:
7899:XOR linked list
7865:
7841:Circular buffer
7822:
7741:
7720:
7718:Data structures
7715:
7685:
7680:
7584:
7463:
7410:
7339:
7335:Weight-balanced
7290:Order statistic
7204:
7196:
7191:
7144:Paul E. Black.
7134:
7130:
7104:10.1145/2689412
7087:
7051:
7049:Further reading
7046:
7039:
7035:
7028:
7004:
7000:
6993:
6969:
6965:
6956:
6954:
6940:
6936:
6921:
6889:
6882:
6867:
6853:
6849:
6838:
6829:
6814:
6796:
6792:
6777:
6763:
6754:
6739:
6725:
6721:
6713:, as coined by
6707:
6703:
6696:
6679:
6668:
6649:
6646:
6645:
6628:
6624:
6622:
6619:
6618:
6602:
6599:
6598:
6582:
6574:
6572:
6569:
6568:
6552:
6549:
6548:
6517:
6499:
6491:
6490:
6484:
6478:
6474:
6469:
6468:
6465:
6444:
6442:
6439:
6438:
6420:
6417:
6416:
6393:
6379:
6376:
6375:
6359:
6356:
6355:
6353:
6349:
6345:
6333:
6327:
6320:
6313:
6286:
6282:
6257:
6253:
6238:
6223:
6219:
6177:
6140:
6139:
6115:
6111:
6102:
6095:
6093:
6090:
6089:
6054:
6048:
6044:
6043:
6037:
6023:
6020:
6019:
5977:
5973:
5961:
5953:
5950:
5949:
5913:
5909:
5908:
5902:
5894:
5891:
5890:
5855:
5848:
5845:
5837:
5834:
5833:
5813:
5812:
5782:
5778:
5767:
5766:
5736:
5732:
5723:
5716:
5714:
5711:
5710:
5698:
5690:
5682:
5678:
5663:
5659:
5654:
5653:
5650:
5647:
5644:
5641:
5638:
5635:
5632:
5629:
5626:
5623:
5620:
5617:
5614:
5611:
5608:
5605:
5602:
5599:
5596:
5593:
5590:
5587:
5584:
5581:
5578:
5575:
5572:
5569:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5545:
5542:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5509:
5506:
5503:
5500:
5497:
5494:
5491:
5488:
5485:
5482:
5479:
5476:
5473:
5470:
5467:
5464:
5461:
5458:
5455:
5452:
5449:
5446:
5443:
5440:
5437:
5434:
5431:
5428:
5425:
5422:
5419:
5416:
5413:
5410:
5407:
5404:
5401:
5398:
5395:
5392:
5389:
5386:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5362:
5359:
5356:
5353:
5350:
5347:
5344:
5341:
5338:
5335:
5332:
5329:
5326:
5323:
5320:
5317:
5314:
5311:
5308:
5305:
5302:
5299:
5296:
5293:
5290:
5287:
5284:
5281:
5278:
5275:
5272:
5269:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5245:
5242:
5239:
5236:
5233:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5209:
5206:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5179:
5176:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5095:
5092:
5089:
5086:
5069:
5065:
5028:
5018:
4993:
4985:
4981:
4977:
4973:
4969:
4965:
4959:
4956:
4954:Double rotation
4951:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4603:
4599:
4557:
4548:
4544:
4533:
4529:
4525:
4521:
4517:
4513:
4507:
4504:
4502:Simple rotation
4488:
4487:
4477:
4467:
4466:
4456:
4445:
4434:
4419:
4407:
4390:(see figure 3)
4385:
4363:
4346:(see figure 2)
4341:
4248:
4244:
4223:
4196:
4193:
4192:
4152:
4150:
4147:
4146:
4110:
4107:
4106:
4090:
4087:
4086:
4049:
4048:
4044:
4034:
4030:
4025:
4023:
4020:
4019:
3993:
3985:non-destructive
3977:
3975:
3971:
3967:
3963:
3959:
3951:
3947:
3943:
3939:
3935:
3928:= nil:
3927:
3920:
3913:= nil:
3912:
3904:
3900:
3888:
3870:
3864:
3860:
3856:
3853:
3847:
3844:
3838:
3835:
3830:
3796:
3780:
3768:
3764:
3760:
3756:
3752:
3749:
3740:
3736:
3732:
3728:
3724:
3720:
3716:
3712:
3708:
3705:
3703:
3699:
3691:
3687:
3679:
3675:
3667:
3663:
3655:
3651:
3643:
3639:
3631:
3629:
3625:
3617:
3600:
3584:
3580:
3572:
3568:
3564:
3552:
3536:
3533:
3527:
3523:
3519:
3516:
3510:
3506:
3503:
3497:
3490:
3484:
3483:is higher than
3481:
3475:
3472:
3466:
3462:
3459:
3453:
3446:
3440:
3437:
3431:
3427:
3423:
3420:
3414:
3411:
3405:
3401:
3398:
3392:
3389:
3383:
3349:
3336:
3332:
3324:
3316:
3303:
3298:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2420:
2386:
2373:
2369:
2361:
2353:
2337:
2332:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1505:
1474:
1470:of each node.
1456:
1440:
1433:
1421:
1400:
1397:
1384:
1378:
1372:
1367:(or at least a
1357:
1345:
1318:
1314:
1305:
1301:
1299:
1296:
1295:
1273:
1266:
1262:
1256:
1252:
1247:
1244:
1243:
1214:
1210:
1208:
1205:
1204:
1188:
1185:
1184:
1142:
1138:
1134:
1121:
1117:
1116:
1114:
1106:
1103:
1102:
1067:
1060:
1057:
1049:
1046:
1045:
1001:
997:
964:
960:
958:
955:
954:
935:
932:
931:
915:
912:
911:
904:
868:
866:
863:
862:
831:
829:
826:
825:
794:
792:
789:
788:
754:
746:
726:
718:
701:
699:
696:
695:
681:
676:
648:
640:
637:
636:
620:
617:
616:
613:weight-balanced
580:
578:
575:
574:
571:redāblack trees
528:
525:
524:
513:
429:
427:
424:
423:
391:
389:
386:
385:
346:
344:
341:
340:
308:
306:
303:
302:
263:
261:
258:
257:
225:
223:
220:
219:
187:Time complexity
156:
154:
151:
150:
82:
75:
64:
58:
55:
48:
44:Manual of Style
36:
32:
23:
22:
15:
12:
11:
5:
8100:
8090:
8089:
8084:
8079:
8074:
8069:
8052:
8051:
8049:
8048:
8042:
8039:
8038:
8036:
8035:
8030:
8025:
8019:
8017:
8011:
8010:
8008:
8007:
8006:
8005:
7995:
7994:
7993:
7991:Hilbert R-tree
7988:
7983:
7973:
7972:
7971:
7969:Fibonacci heap
7966:
7961:
7951:
7950:
7949:
7944:
7939:
7937:Redāblack tree
7934:
7929:
7919:
7913:
7911:
7905:
7904:
7902:
7901:
7896:
7891:
7886:
7881:
7875:
7873:
7867:
7866:
7864:
7863:
7858:
7853:
7848:
7843:
7838:
7832:
7830:
7824:
7823:
7821:
7820:
7819:
7818:
7813:
7803:
7802:
7801:
7794:Priority queue
7791:
7790:
7789:
7779:
7774:
7769:
7768:
7767:
7762:
7751:
7749:
7743:
7742:
7740:
7739:
7734:
7728:
7726:
7722:
7721:
7714:
7713:
7706:
7699:
7691:
7682:
7681:
7679:
7678:
7673:
7668:
7663:
7658:
7653:
7648:
7643:
7638:
7633:
7628:
7623:
7618:
7613:
7608:
7603:
7598:
7592:
7590:
7586:
7585:
7583:
7582:
7577:
7572:
7567:
7562:
7557:
7552:
7547:
7542:
7537:
7532:
7527:
7522:
7517:
7500:
7495:
7490:
7485:
7480:
7474:
7472:
7465:
7464:
7462:
7461:
7456:
7451:
7449:Ternary search
7446:
7441:
7436:
7431:
7426:
7420:
7418:
7412:
7411:
7409:
7408:
7403:
7398:
7393:
7388:
7383:
7378:
7373:
7365:
7360:
7355:
7349:
7347:
7341:
7340:
7338:
7337:
7332:
7327:
7322:
7317:
7312:
7307:
7297:
7292:
7287:
7282:
7277:
7272:
7262:
7257:
7252:
7247:
7242:
7237:
7232:
7227:
7222:
7216:
7214:
7198:
7197:
7190:
7189:
7182:
7175:
7167:
7161:
7160:
7129:
7128:External links
7126:
7125:
7124:
7076:
7050:
7047:
7045:
7044:
7033:
7026:
6998:
6991:
6963:
6934:
6919:
6880:
6865:
6847:
6827:
6812:
6790:
6775:
6752:
6737:
6719:
6701:
6694:
6666:
6653:
6631:
6627:
6606:
6585:
6581:
6577:
6556:
6546:
6545:
6534:
6531:
6525:
6522:
6516:
6509:
6506:
6502:
6498:
6494:
6487:
6481:
6477:
6472:
6464:
6461:
6458:
6452:
6449:
6424:
6401:
6398:
6392:
6389:
6386:
6383:
6363:
6343:
6318:
6311:
6280:
6265:(in Russian).
6251:
6220:
6218:
6215:
6214:
6213:
6208:
6203:
6198:
6196:Scapegoat tree
6193:
6188:
6183:
6176:
6173:
6169:geometric mean
6157:
6156:
6155:
6154:
6138:
6135:
6132:
6129:
6126:
6123:
6118:
6114:
6110:
6106:
6103:
6101:
6098:
6097:
6083:
6082:
6070:
6067:
6058:
6051:
6047:
6042:
6036:
6033:
6030:
6027:
6007:
6004:
6001:
5997:
5994:
5991:
5988:
5985:
5980:
5976:
5969:
5966:
5960:
5957:
5937:
5934:
5931:
5924:
5921:
5916:
5912:
5907:
5901:
5898:
5874:
5871:
5865:
5859:
5854:
5851:
5844:
5841:
5829:
5828:
5827:
5826:
5811:
5808:
5805:
5802:
5799:
5796:
5793:
5790:
5785:
5781:
5777:
5773:
5770:
5768:
5765:
5762:
5759:
5756:
5753:
5750:
5747:
5744:
5739:
5735:
5731:
5727:
5724:
5722:
5719:
5718:
5694:height limit.
5672:tail-recursive
5658:
5655:
5085:
5081:
5080:
5077:
5073:
5072:
5062:
5059:
5058:
5055:
5052:
5051:
5048:
5042:
5041:
4991:
4983:
4979:
4975:
4971:
4967:
4963:
4955:
4952:
4619:
4615:
4614:
4611:
4607:
4606:
4596:
4593:
4592:
4589:
4586:
4585:
4582:
4576:
4575:
4546:
4542:
4531:
4527:
4523:
4519:
4515:
4511:
4503:
4500:
4414:
4413:
4410:
4404:
4401:
4398:
4395:
4392:
4391:
4388:
4382:
4379:
4376:
4373:
4370:
4369:
4366:
4360:
4357:
4354:
4351:
4348:
4347:
4344:
4338:
4335:
4332:
4329:
4319:
4318:
4315:
4309:
4306:
4303:
4302:
4299:
4293:
4290:
4287:
4286:
4283:
4277:
4274:
4271:
4270:
4267:
4261:
4258:
4246:
4242:
4228:tree rotations
4222:
4219:
4206:
4203:
4200:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4144:parallel depth
4127:
4124:
4121:
4118:
4114:
4094:
4073:
4068:
4064:
4061:
4056:
4053:
4047:
4043:
4040:
4037:
4033:
3990:
3989:
3973:
3969:
3965:
3961:
3957:
3949:
3945:
3941:
3937:
3933:
3925:
3918:
3910:
3902:
3898:
3893:
3890:
3889:
3886:
3881:
3851:
3842:
3832:
3831:
3801:
3798:
3797:
3794:
3789:
3746:
3745:
3715:'s left child
3701:
3697:
3689:
3685:
3677:
3673:
3665:
3661:
3660:JoinRightAVL(T
3653:
3649:
3641:
3637:
3632:
3627:
3623:
3618:
3598:
3582:
3578:
3570:
3566:
3562:
3561:JoinRightAVL(T
3557:
3554:
3553:
3550:
3545:
3531:
3514:
3501:
3488:
3479:
3470:
3457:
3444:
3435:
3426:. It requires
3418:
3409:
3396:
3387:
3361:set difference
3348:
3345:
3300:
3299:
2426:
2422:
2421:
2418:
2413:
2409:
2408:
2385:
2382:
2334:
2333:
1511:
1507:
1506:
1503:
1498:
1494:
1493:
1468:balance factor
1455:
1452:
1396:
1393:
1369:total preorder
1356:
1353:
1344:
1341:
1329:
1326:
1321:
1317:
1313:
1308:
1304:
1276:
1272:
1269:
1265:
1259:
1255:
1251:
1231:
1228:
1223:
1220:
1217:
1213:
1192:
1172:
1169:
1165:
1162:
1159:
1153:
1150:
1145:
1141:
1137:
1132:
1129:
1124:
1120:
1113:
1110:
1097: is the
1086:
1083:
1077:
1071:
1066:
1063:
1056:
1053:
1042:
1041:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1004:
1000:
996:
993:
990:
987:
984:
981:
978:
975:
972:
967:
963:
939:
919:
903:
900:
887:
884:
881:
878:
875:
850:
847:
844:
841:
838:
813:
810:
807:
804:
801:
787:A node X with
782:
781:
770:
767:
764:
761:
753:
745:
742:
739:
736:
733:
725:
717:
714:
711:
708:
689:balance factor
680:
679:Balance factor
677:
675:
672:
656:
653:
647:
644:
624:
599:
596:
593:
590:
587:
564:data structure
560:Evgenii Landis
545:tree rotations
532:
464:
463:
460:
459:
448:
445:
442:
439:
436:
421:
410:
407:
404:
401:
398:
383:
381:
377:
376:
365:
362:
359:
356:
353:
338:
327:
324:
321:
318:
315:
300:
298:
294:
293:
282:
279:
276:
273:
270:
255:
244:
241:
238:
235:
232:
217:
215:
211:
210:
203:
196:
194:
190:
189:
183:
182:
180:
169:
166:
163:
148:
146:
142:
141:
131:
130:
128:big O notation
123:
122:
120:Evgenii Landis
113:
109:
108:
105:
101:
100:
95:
89:
88:
80:
77:
76:
39:
37:
30:
9:
6:
4:
3:
2:
8099:
8088:
8085:
8083:
8080:
8078:
8075:
8073:
8070:
8068:
8065:
8064:
8062:
8047:
8044:
8043:
8040:
8034:
8031:
8029:
8026:
8024:
8021:
8020:
8018:
8016:
8012:
8004:
8001:
8000:
7999:
7996:
7992:
7989:
7987:
7984:
7982:
7979:
7978:
7977:
7974:
7970:
7967:
7965:
7964:Binomial heap
7962:
7960:
7957:
7956:
7955:
7952:
7948:
7945:
7943:
7940:
7938:
7935:
7933:
7930:
7928:
7925:
7924:
7923:
7920:
7918:
7915:
7914:
7912:
7910:
7906:
7900:
7897:
7895:
7892:
7890:
7887:
7885:
7882:
7880:
7877:
7876:
7874:
7872:
7868:
7862:
7861:Sparse matrix
7859:
7857:
7854:
7852:
7849:
7847:
7846:Dynamic array
7844:
7842:
7839:
7837:
7834:
7833:
7831:
7829:
7825:
7817:
7814:
7812:
7809:
7808:
7807:
7804:
7800:
7797:
7796:
7795:
7792:
7788:
7785:
7784:
7783:
7780:
7778:
7775:
7773:
7770:
7766:
7763:
7761:
7758:
7757:
7756:
7753:
7752:
7750:
7748:
7744:
7738:
7735:
7733:
7730:
7729:
7727:
7723:
7719:
7712:
7707:
7705:
7700:
7698:
7693:
7692:
7689:
7677:
7674:
7672:
7669:
7667:
7664:
7662:
7659:
7657:
7654:
7652:
7649:
7647:
7644:
7642:
7639:
7637:
7634:
7632:
7629:
7627:
7626:Hash calendar
7624:
7622:
7619:
7617:
7614:
7612:
7609:
7607:
7604:
7602:
7599:
7597:
7594:
7593:
7591:
7587:
7581:
7578:
7576:
7573:
7571:
7568:
7566:
7563:
7561:
7558:
7556:
7553:
7551:
7548:
7546:
7543:
7541:
7538:
7536:
7533:
7531:
7528:
7526:
7523:
7521:
7518:
7515:
7513:
7507:
7505:
7501:
7499:
7496:
7494:
7491:
7489:
7486:
7484:
7481:
7479:
7476:
7475:
7473:
7470:
7466:
7460:
7457:
7455:
7452:
7450:
7447:
7445:
7442:
7440:
7437:
7435:
7432:
7430:
7427:
7425:
7422:
7421:
7419:
7417:
7413:
7407:
7404:
7402:
7401:van Emde Boas
7399:
7397:
7394:
7392:
7391:Skew binomial
7389:
7387:
7384:
7382:
7379:
7377:
7374:
7372:
7370:
7366:
7364:
7361:
7359:
7356:
7354:
7351:
7350:
7348:
7346:
7342:
7336:
7333:
7331:
7328:
7326:
7323:
7321:
7318:
7316:
7313:
7311:
7308:
7306:
7302:
7298:
7296:
7293:
7291:
7288:
7286:
7283:
7281:
7278:
7276:
7273:
7271:
7270:Binary search
7267:
7263:
7261:
7258:
7256:
7253:
7251:
7248:
7246:
7243:
7241:
7238:
7236:
7233:
7231:
7228:
7226:
7223:
7221:
7218:
7217:
7215:
7212:
7208:
7203:
7199:
7195:
7188:
7183:
7181:
7176:
7174:
7169:
7168:
7165:
7157:
7153:
7152:
7147:
7141:
7132:
7131:
7121:
7117:
7113:
7109:
7105:
7101:
7097:
7093:
7086:
7082:
7077:
7074:
7073:0-201-89685-0
7070:
7066:
7062:
7061:
7056:
7053:
7052:
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3792:
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3774:
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3695:
3684:JoinLeftAVL(T
3683:
3676:)>Height(T
3671:
3659:
3652:)>Height(T
3647:
3635:
3622:JoinLeftAVL(T
3621:
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3604:
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3592:
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3542:
3530:
3513:
3505:until a node
3500:
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3487:
3478:
3469:
3456:
3451:
3443:
3434:
3417:
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3395:
3386:
3381:
3378:The function
3376:
3374:
3370:
3366:
3362:
3358:
3354:
3344:
3340:
3328:
3320:
3313:
3309:
3306:
2424:
2423:
2417:
2416:
2412:
2406:
2405:
2404:
2402:
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2393:
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2365:
2357:
2350:
2347:
2344:
2341:
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1508:
1502:
1501:
1497:
1491:
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1487:
1486:
1483:
1481:
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1469:
1463:
1461:
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1444:
1436:
1428:
1424:
1419:
1415:
1411:
1406:
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1388:
1381:
1375:
1370:
1366:
1362:
1352:
1350:
1340:
1327:
1324:
1319:
1315:
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1302:
1293:
1270:
1267:
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1253:
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1215:
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953:
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715:
709:
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690:
686:
671:
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645:
642:
622:
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594:
591:
588:
572:
567:
565:
561:
557:
553:
548:
546:
530:
520:
516:
511:
507:
503:
499:
495:
491:
487:
478:
470:
443:
440:
437:
422:
405:
402:
399:
384:
382:
378:
360:
357:
354:
339:
322:
319:
316:
301:
299:
295:
277:
274:
271:
256:
239:
236:
233:
218:
216:
212:
209:
208:
204:
202:
201:
197:
195:
191:
188:
184:
181:
164:
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147:
143:
140:
136:
132:
129:
124:
121:
117:
114:
110:
106:
102:
99:
96:
94:
90:
85:
73:
70:
62:
59:November 2021
52:
47:
45:
38:
29:
28:
19:
8082:Search trees
8072:Binary trees
7931:
7816:Disjoint-set
7511:
7503:
7368:
7301:Left-leaning
7239:
7207:dynamic sets
7202:Search trees
7149:
7095:
7091:
7064:
7063:, Volume 3:
7058:
7055:Donald Knuth
7036:
7008:
7001:
6973:
6966:
6955:. Retrieved
6947:
6937:
6892:
6856:
6850:
6841:
6801:
6793:
6766:
6728:
6722:
6704:
6685:
6346:
6298:
6283:
6275:
6266:
6260:
6254:
6246:the original
6158:
5887:golden ratio
5699:
5696:
5684:
5665:
5660:
5033:
5029:
5024:
5021:rotate_Right
5020:
5014:
5010:
5006:
4996:
4988:
4957:
4566:
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4558:
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4364:rotate_Right
4322:
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4280:
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4240:
4232:
4224:
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4008:
4004:
4000:
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3994:
3980:
3978:
3953:
3929:
3921:
3914:
3906:
3894:
3875:
3871:
3865:
3863:, is an AVL
3848:
3839:
3836:
3826:
3822:
3818:
3814:
3810:
3806:
3802:
3782:
3776:
3772:
3750:
3706:
3693:
3681:
3669:
3657:
3645:
3633:
3619:
3613:
3610:
3609:T''
3606:
3602:
3594:
3590:
3586:
3574:
3558:
3540:
3528:
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3498:
3493:
3485:
3476:
3467:
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3432:
3415:
3406:
3393:
3384:
3379:
3377:
3372:
3368:
3364:
3363:. Then fast
3357:intersection
3350:
3338:
3326:
3318:
3314:
3310:
3307:
3304:
2971:rotate_Right
2410:
2397:
2394:
2387:
2375:
2363:
2355:
2351:
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2345:
2342:
2338:
2023:rotate_Right
1495:
1484:
1472:
1464:
1457:
1446:
1442:
1434:
1426:
1422:
1413:
1409:
1407:
1401:
1398:
1386:
1379:
1373:
1358:
1346:
1242:nodes where
1099:golden ratio
1043:
909:
905:
786:
783:
728:RightSubtree
688:
682:
568:
549:
518:
504:andis) is a
501:
497:
493:
489:
483:
206:
199:
65:
56:
49:Please help
41:
7959:Binary heap
7884:Linked list
7601:Exponential
7589:Other trees
6242:"AVL Trees"
6167:ā0.947 and
5885: the
5291:right_child
5231:right_child
5159:right_child
5030:rotate_Left
5027:followed by
4998:decreases.
4684:right_child
4627:rotate_Left
4559:rotate_Left
4534:may occur.
4342:rotate_Left
4331:Right Right
4260:Right Right
4221:Rebalancing
4140:in parallel
3952:.root)
3944:) = Split(t
3422:as well as
3214:right_child
2674:rotate_Left
2581:right_child
2401:Rebalancing
2269:right_child
1735:rotate_Left
1588:right_child
1365:total order
910:The height
756:LeftSubtree
685:binary tree
554:inventors,
112:Invented by
8061:Categories
7947:Splay tree
7851:Hash table
7732:Collection
7545:Priority R
7295:Palindrome
7146:"AVL Tree"
6957:2016-07-02
6944:"AVL tree"
6902:1602.02120
6813:0071378707
6547:holds and
6299:Algorithms
6269:: 263ā266.
6217:References
6191:Splay tree
6161:asymptotic
6018:and
5351:left_child
5279:left_child
5171:left_child
5135:left_child
4744:left_child
4669:left_child
4449: := ā
4397:Left Right
4375:Right Left
4308:Left Right
4292:Right Left
3719:, the key
3400:and a key
3190:left_child
3178:left_child
2878:left_child
2518:left_child
2245:left_child
2233:left_child
1482:the tree.
1480:rebalances
1343:Operations
902:Properties
674:Definition
500:elsky and
207:Worst Case
8003:Hash tree
7889:Skip list
7836:Bit array
7737:Container
7631:iDistance
7510:implicit
7498:Hilbert R
7493:Cartesian
7376:Fibonacci
7310:Scapegoat
7305:Redāblack
6785:312435417
6747:939446542
6555:μ
6533:μ
6515:≤
6463:≤
6460:μ
6457:−
6391:≤
6388:μ
6385:≤
6362:μ
6181:WAVL tree
6122:
6105:≦
6066:≈
6046:φ
6000:−
5996:≈
5990:−
5984:
5930:≈
5923:φ
5920:
5870:≈
5840:φ
5789:
5743:
5726:≦
5676:amortized
4481: !=
4406:rotation
4384:rotation
4362:rotation
4353:Left Left
4340:rotation
4311:ā¹ Z is a
4295:ā¹ Z is a
4279:ā¹ Z is a
4276:Left Left
4263:ā¹ Z is a
4173:
4164:
4120:≥
4042:
3672:(Height(T
3648:(Height(T
1490:Invariant
1418:amortized
1395:Traversal
1355:Searching
1271:∈
1227:−
1168:−
1164:≈
1158:−
1152:φ
1149:
1128:
1082:≈
1052:φ
1008:
1003:φ
989:≤
971:
744:−
646:≤
643:μ
623:μ
592:
441:
403:
358:
320:
275:
237:
200:Amortized
18:AVL trees
7932:AVL tree
7811:Multiset
7760:Multimap
7747:Abstract
7646:Link/cut
7358:Binomial
7285:Interval
7083:(2015),
6875:61278554
6822:48139308
6684:(2000).
6291:(1983).
6175:See also
6171:ā0.910.
3895:function
3803:function
3634:function
3620:function
3559:function
3106:continue
2812:continue
2158:continue
1873:continue
1414:previous
490:AVL tree
193:Function
104:Invented
87:AVL tree
7986:R+ tree
7981:R* tree
7927:AA tree
7606:Fenwick
7570:Segment
7469:Spatial
7386:Pairing
7381:Leftist
7303:)
7275:Dancing
7268:)
7266:Optimal
7120:1407290
7112:3361215
6929:2897793
6153: .
5948:
5076:Result:
5032:around
5023:around
4610:Result:
4489:rotate_
4468:rotate_
4444:} with
4191:. When
4142:with a
3905:):
3897:Union(t
3522:, root
3461:, root
1290:is the
1171:0.3277.
496:delson-
8015:Graphs
7976:R-tree
7917:B-tree
7871:Linked
7828:Arrays
7656:Merkle
7621:Fusion
7611:Finger
7535:Octree
7525:Metric
7459:Y-fast
7454:X-fast
7444:Suffix
7363:Brodal
7353:Binary
7118:
7110:
7071:
7024:
6989:
6927:
6917:
6873:
6863:
6820:
6810:
6783:
6773:
6745:
6735:
6692:
6309:
6206:T-tree
6201:B-tree
6165:median
5832:where
5683:O(log
5664:O(log
5639:return
5372:parent
5330:parent
5252:parent
5210:parent
5047:Input:
4936:return
4765:parent
4723:parent
4581:Input:
4425:where
4403:double
4381:double
4359:simple
4337:simple
4007:calls
3979:Here,
3954:return
3940:, b, t
3930:return
3915:return
3827:return
3819:return
3781:O(log
3735:, key
3700:, k, T
3696:Node(T
3694:return
3692:)
3688:, k, T
3682:return
3668:)
3664:, k, T
3658:return
3644:)
3640:, k, T
3636:Join(T
3626:, k, T
3614:return
3607:return
3591:return
3587:return
3573:)
3565:, k, T
3343:time.
3337:O(log
3325:O(log
3317:O(log
3124:parent
2488:parent
2440:parent
2384:Delete
2380:time.
2374:O(log
2362:O(log
2354:O(log
2179:parent
1939:parent
1651:parent
1555:parent
1525:parent
1454:Insert
1425:ā log(
1385:O(log
1044:where
748:Height
720:Height
552:Soviet
380:Delete
297:Insert
214:Search
7909:Trees
7782:Queue
7777:Stack
7725:Types
7666:Range
7636:K-ary
7596:Cover
7439:Radix
7424:Ctrie
7416:Tries
7345:Heaps
7325:Treap
7315:Splay
7280:HTree
7235:(a,b)
7225:2ā3ā4
7142:from
7116:S2CID
7088:(PDF)
6925:S2CID
6897:arXiv
6334:(PDF)
6069:1.065
6003:0.328
5933:1.440
5873:1.618
4982:and t
4545:and t
4447:Right
4441:Right
4297:right
4265:right
4005:Split
3997:Join2
3981:Split
3777:Split
3680:)+1)
3656:)+1)
3369:Split
3353:union
3280:break
3244:->
3058:break
2764:break
2317:break
2299:->
2110:break
1822:break
1085:1.618
683:In a
517:(log
510:child
488:, an
145:Space
7998:Trie
7954:Heap
7772:List
7671:SPQR
7550:Quad
7478:Ball
7434:Hash
7406:Weak
7396:Skew
7371:-ary
7156:NIST
7069:ISBN
7022:ISBN
6987:ISBN
6915:ISBN
6871:OCLC
6861:ISBN
6818:OCLC
6808:ISBN
6781:OCLC
6771:ISBN
6743:OCLC
6733:ISBN
6690:ISBN
6307:ISBN
5772:<
5691:O(1)
5679:O(1)
5558:else
5492:>
5471:else
5324:null
5204:null
5111:node
5099:node
5087:node
5070:))+2
4970:or t
4882:else
4717:null
4645:node
4633:node
4621:node
4604:))+2
4451:Left
4437:Left
4313:left
4281:left
4105:and
4009:Join
4001:Join
3974:>
3972:), t
3964:), t
3962:<
3960:), t
3942:>
3938:<
3859:and
3846:and
3773:Join
3611:else
3595:else
3541:Join
3494:Join
3450:Join
3391:and
3380:Join
3373:Join
3371:and
3365:bulk
3359:and
3333:O(1)
3247:root
3241:tree
3238:else
3211:else
3157:null
2998:else
2959:else
2920:>
2851:<
2824:else
2701:else
2662:else
2623:<
2554:>
2458:null
2370:O(1)
2302:root
2296:tree
2293:else
2266:else
2212:null
2074:>
2050:else
2011:else
1972:>
1912:<
1885:else
1786:<
1762:else
1723:else
1684:<
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