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Indeed, every AVL tree can be colored red–black, [14] but there are RB trees which are not AVL balanced. For maintaining the AVL (or RB) tree's invariants, rotations play an important role. In the worst case, even without rotations, AVL or RB insertions or deletions require O(log n) inspections and/or updates to AVL balance factors (or RB ...
Tree rotations are used in a number of tree data structures such as AVL trees, red–black trees, WAVL trees, splay trees, and treaps. They require only constant time because they are local transformations: they only operate on 5 nodes, and need not examine the rest of the tree.
Binary tree rotations. Operations that modify the tree must make sure that the weight of the left and right subtrees of every node remain within some factor α of each other, using the same rebalancing operations used in AVL trees: rotations and double rotations. Formally, node balance is defined as follows:
The worst-case height of AVL is 0.720 times the worst-case height of red-black trees, so AVL trees are more rigidly balanced. The performance measurements of Ben Pfaff with realistic test cases in 79 runs find AVL to RB ratios between 0.677 and 1.077, median at 0.947, and geometric mean 0.910. [22]
Self-balancing binary trees solve this problem by performing transformations on the tree (such as tree rotations) at key insertion times, in order to keep the height proportional to log 2 (n). Although a certain overhead is involved, it is not bigger than the always necessary lookup cost and may be justified by ensuring fast execution of all ...
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All of the red-black tree algorithms that have been proposed are characterized by a worst-case search time bounded by a small constant multiple of log N in a tree of N keys, and the behavior observed in practice is typically that same multiple faster than the worst-case bound, close to the optimal log N nodes examined that would be observed in a perfectly balanced tree.
Balancing a k-d tree requires care because k-d trees are sorted in multiple dimensions, so the tree-rotation technique cannot be used to balance them as this may break the invariant. Several variants of balanced k-d trees exist. They include divided k-d tree, pseudo k-d tree, K-D-B-tree, hB-tree and Bkd-tree.