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Fig. 1: AVL tree with balance factors (green) In computer science, an AVL tree (named after inventors Adelson-Velsky and Landis) is a self-balancing binary search tree. In an AVL tree, the heights of the two child 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.
The height of an external node is zero, and the height of any internal node is always one plus the maximum of the heights of its two children. Thus, the height function of an AVL tree obeys the constraints of a WAVL tree, and we may convert any AVL tree into a WAVL tree by using the height of each node as its rank. [1] [2]
Most operations on a binary search tree (BST) take time directly proportional to the height of the tree, so it is desirable to keep the height small. A binary tree with height h can contain at most 2 0 +2 1 +···+2 h = 2 h+1 −1 nodes. It follows that for any tree with n nodes and height h: + And that implies:
The black height of a red–black tree is the number of black nodes in any path from the root to the leaves, which, by requirement 4, is constant (alternatively, it could be defined as the black depth of any leaf node). [16]: 154–165 The black height of a node is the black height of the subtree rooted by it. In this article, the black height ...
The height of the root is the height of the tree. The depth of a node is the length of the path to its root (i.e., its root path). Thus the root node has depth zero, leaf nodes have height zero, and a tree with only a single node (hence both a root and leaf) has depth and height zero. Conventionally, an empty tree (tree with no nodes, if such ...
This means that for every LLRB tree, there is a unique corresponding 2–3–4 tree, and vice versa. If we impose the additional requirement that a node may not have two red children, LLRB trees become isomorphic to 2–3 trees, since 4-nodes are now prohibited. Sedgewick remarks that the implementations of LLRB 2–3 trees and LLRB 2–3–4 ...
The size of an internal node is the sum of sizes of its two children, plus one: (size[n] = size[n.left] + size[n.right] + 1). Based on the size, one defines the weight to be weight[n] = size[n] + 1. [a] Weight has the advantage that the weight of a node is simply the sum of the weights of its left and right children. Binary tree rotations.
Height - Length of the path from the root to the deepest node in the tree. A (rooted) tree with only one node (the root) has a height of zero. In the example diagram, the tree has height of 2. Sibling - Nodes that share the same parent node. A node p is an ancestor of a node q if it exists on the path from q to the root. The node q is then ...