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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 n 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.
The weak AVL tree is defined by the weak AVL rule: Weak AVL rule: all rank differences are 1 or 2, and all leaf nodes have rank 0. Note that weak AVL tree generalizes the AVL tree by allowing for 2,2 type node. A simple proof shows that a weak AVL tree can be colored in a way that represents a red-black tree.
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] The performance of WAVL trees lie in between ...
This unsorted tree has non-unique values (e.g., the value 2 existing in different nodes, not in a single node only) and is non-binary (only up to two children nodes per parent node in a binary tree). The root node at the top (with the value 2 here), has no parent as it is the highest in the tree hierarchy.
The depth of a tree is the maximum depth of any vertex. Depth is commonly needed in the manipulation of the various self-balancing trees, AVL trees in particular. The root has depth zero, leaves have height zero, and a tree with only a single vertex (hence both a root and leaf) has depth and height zero.
Various height-balanced binary search trees were introduced to confine the tree height, such as AVL trees, Treaps, and red–black trees. [5] The AVL tree was invented by Georgy Adelson-Velsky and Evgenii Landis in 1962 for the efficient organization of information. [6] [7] It was the first self-balancing binary search tree to be invented. [8]
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.
For comparison, an AVL tree is guaranteed to be within a factor of 1.44 of the optimal height while requiring only two additional bits of storage in a naive implementation. [1] Therefore, most self-balancing BST algorithms keep the height within a constant factor of this lower bound.