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In computing, a threaded binary tree is a binary tree variant that facilitates traversal in a particular order. An entire binary search tree can be easily traversed in order of the main key, but given only a pointer to a node, finding the node which comes next may be slow or impossible. For example, leaf nodes by definition have no descendants ...
For infinite trees, simple algorithms often fail this. For example, given a binary tree of infinite depth, a depth-first search will go down one side (by convention the left side) of the tree, never visiting the rest, and indeed an in-order or post-order traversal will never visit any nodes, as it has not reached a leaf (and in fact never will ...
threaded binary tree; threaded tree; three-dimensional; three-way merge sort; three-way radix quicksort; time-constructible function; time/space complexity; top-down radix sort; top-down tree automaton; top-node; topological order; topological sort; topology tree; total function; totally decidable language; totally decidable problem; totally ...
In addition to its dynamic programming algorithm, Knuth proposed two heuristics (or rules) to produce nearly (approximation of) optimal binary search trees. Studying nearly optimal binary search trees was necessary since Knuth's algorithm time and space complexity can be prohibitive when is substantially large. [6]
For all nodes, the left subtree's key must be less than the node's key, and the right subtree's key must be greater than the node's key. These subtrees must all qualify as binary search trees. The worst-case time complexity for searching a binary search tree is the height of the tree, which can be as small as O(log n) for a tree with n elements.
In this case, an advantage of using a binary tree is significantly reduced because it is essentially a linked list which time complexity is O(n) (n as the number of nodes) and it has more data space than the linked list due to two pointers per node, while the complexity of O(log 2 n) for data search in a balanced binary tree is normally expected.
An algorithm is said to be constant time (also written as () time) if the value of () (the complexity of the algorithm) is bounded by a value that does not depend on the size of the input. For example, accessing any single element in an array takes constant time as only one operation has to be performed to locate 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.