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A tournament tree can be represented as a balanced binary tree by adding sentinels to the input lists (i.e. adding a member to the end of each list with a value of infinity) and by adding null lists (comprising only a sentinel) until the number of lists is a power of two. The balanced tree can be stored in a single array.
A binary tree is a rooted tree that is also an ordered tree (a.k.a. plane tree) in which every node has at most two children. A rooted tree naturally imparts a notion of levels (distance from the root); thus, for every node, a notion of children may be defined as the nodes connected to it a level below.
A node is α-weight-balanced if weight[n.left] ≥ α·weight[n] and weight[n.right] ≥ α·weight[n]. [7] Here, α is a numerical parameter to be determined when implementing weight balanced trees. Larger values of α produce "more balanced" trees, but not all values of α are appropriate; Nievergelt and Reingold proved that
In computer science, tree traversal (also known as tree search and walking the tree) is a form of graph traversal and refers to the process of visiting (e.g. retrieving, updating, or deleting) each node in a tree data structure, exactly once. Such traversals are classified by the order in which the nodes are visited.
In computer science, an optimal binary search tree (Optimal BST), sometimes called a weight-balanced binary tree, [1] is a binary search tree which provides the smallest possible search time (or expected search time) for a given sequence of accesses (or access probabilities). Optimal BSTs are generally divided into two types: static and dynamic.
In 2016, Blelloch et al. formally proposed the join-based algorithms, and formalized the join algorithm for four different balancing schemes: AVL trees, red–black trees, weight-balanced trees and treaps. In the same work they proved that Adams' algorithms on union, intersection and difference are work-optimal on all the four balancing schemes.
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.
Removing a point from a balanced k-d tree takes O(log n) time. Querying an axis-parallel range in a balanced k-d tree takes O(n 1−1/k +m) time, where m is the number of the reported points, and k the dimension of the k-d tree. Finding 1 nearest neighbour in a balanced k-d tree with randomly distributed points takes O(log n) time on average.