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For height-balanced binary trees, the height is defined to be logarithmic () in the number of items. This is the case for many binary search trees, such as AVL trees and red–black trees . Splay trees and treaps are self-balancing but not height-balanced, as their height is not guaranteed to be logarithmic in the number of items.
For lookup-intensive applications, AVL trees are faster than red–black trees because they are more strictly balanced. [4] Similar to red–black trees, AVL trees are height-balanced. Both are, in general, neither weight-balanced nor -balanced for any ; [5] that is, sibling nodes can have hugely differing numbers of descendants.
The number of different binary trees on nodes is , the th Catalan number (assuming we view trees with identical structure as identical). For large n {\displaystyle n} , this is about 4 n {\displaystyle 4^{n}} ; thus we need at least about log 2 4 n = 2 n {\displaystyle \log _{2}4^{n}=2n} bits to encode it.
Fig. 1: A binary search tree of size 9 and depth 3, with 8 at the root. In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree.
Its balance function needed to take care of only 4 unbalanced cases and one default balanced case. [11] The original algorithm used 8 unbalanced cases, but Cormen et al. (2001) reduced that to 6 unbalanced cases. [1] Sedgewick showed that the insert operation can be implemented in just 46 lines of Java code.
Join: The function Join is on two weight-balanced trees t 1 and t 2 and a key k and will return a tree containing all elements in t 1, t 2 as well as k. It requires k to be greater than all keys in t 1 and smaller than all keys in t 2. If the two trees have the balanced weight, Join simply create a new node with left subtree t 1, root k and ...
As with other trees, B-trees can be represented as a collection of three types of nodes: root, internal (a.k.a. interior), and leaf. Note the following variable definitions: K: Maximum number of potential search keys for each node in a B-tree. (this value is constant over the entire tree).
WAVL trees are designed to combine some of the best properties of both AVL trees and red–black trees. One advantage of AVL trees over red–black trees is being more balanced: they have height at most (for a tree with n data items, where is the golden ratio), while red–black trees have larger maximum height, . If a WAVL tree is ...