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In computing, a threaded binary tree is a binary tree variant that facilitates traversal in a particular order (often the same order already defined for the tree).

## Contents

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, so no other node can be reached given only a pointer to a leaf node -- of course that includes the desired "next" node. A threaded tree adds extra information in some or all nodes, so the "next" node can be found quickly. It can also be traversed without recursion and the extra storage (proportional to the tree's depth) that requires.

"A binary tree is threaded by making all right child pointers that would normally be null point to the in-order successor of the node (if it exists), and all left child pointers that would normally be null point to the in-order predecessor of the node." 

This assumes the traversal order is the same as in-order traversal of the tree. However, pointers can instead (or in addition) be added to tree nodes, rather than replacing. Linked lists thus defined are also commonly called "threads", and can be used to enable traversal in any order(s) desired. For example, a tree whose nodes represent information about people might be sorted by name, but have extra threads allowing quick traversal in order of birth date, weight, or any other known characteristic.

## Motivation

Trees, including (but not limited to) binary search trees, can be used to store items in a particular order, such as the value of some property stored in each node, often called a key. One useful operation on such a tree is traversal: visiting all the items in order of the key.

A simple recursive traversal algorithm that visits each node of a binary search tree is the following. Assume t is a pointer to a node, or nil. "Visiting" t can mean performing any action on the node t or its contents.

Algorithm traverse(t):

• Input: a pointer t to a node (or nil)
• If t = nil, return.
• Else:
• traverse(left-child(t))
• Visit t
• traverse(right-child(t))

One problem with this algorithm is that, because of its recursion, it uses stack space proportional to the height of a tree. If the tree is fairly balanced, this amounts to O(log n) space for a tree containing n elements. In the worst case, when the tree takes the form of a chain, the height of the tree is n so the algorithm takes O(n) space. A second problem is that all traversals must begin at the root when nodes have pointers only to their children. It is common to have a pointer to a particular node, but that is not sufficient to get back to the rest of the tree unless extra information is added, such as thread pointers.

In this approach, it may not be possible to tell whether the left and/or right pointers in a given node actually point to children, or are a consequence of threading. If the distinction is necessary, adding a single bit to each node is enough to record it.

In a 1968 textbook, Donald Knuth asked whether a non-recursive algorithm for in-order traversal exists, that uses no stack and leaves the tree unmodified.  One of the solutions to this problem is tree threading, presented by J. H. Morris in 1979.   In the 1969 follow-up edition,  Knuth attributed the threaded tree representation to Perlis and Thornton (1960). 

### Relation to parent pointers

Another way to achieve similar goals is to include a pointer in every node, to that node's parent node. Given that, the "next" node can always be reached. "right" pointers are still null whenever there are no right children. To find the "next" node from a node whose right pointer is null, walk up through "parent" pointers until reaching a node whose right pointer is not null, and is not the child you just came up from. That node is the "next" node, and after it come its descendants on the right.

It is also possible to discover the parent of a node from a threaded binary tree, without explicit use of parent pointers or a stack, although it is slower. To see this, consider a node k with right child r. Then the left pointer of r must be either a child or a thread back to k. In the case that r has a left child, that left child must in turn have either a left child of its own or a thread back to k, and so on for all successive left children. So by following the chain of left pointers from r, we will eventually find a thread pointing back to k. The situation is symmetrically similar when q is the left child of pwe can follow q's right children to a thread pointing ahead to p.

## Types

1. Single Threaded: each node is threaded towards either the in-order predecessor or successor (left or right).
2. Double threaded: each node is threaded towards both the in-order predecessor and successor (left and right).

In Python:

`defparent(node):ifnodeisnode.tree.root:returnNoneelse:x=nodey=nodewhileTrue:ifis_thread(y):p=y.rightifpisNoneorp.leftisnotnode:p=xwhilenotis_thread(p.left):p=p.leftp=p.leftreturnpelifis_thread(x):p=x.leftifpisNoneorp.rightisnotnode:p=ywhilenotis_thread(p.right):p=p.rightp=p.rightreturnpx=x.lefty=y.right`

## The array of in-order traversal

Threads are reference to the predecessors and successors of the node according to an inorder traversal.

In-order traversal of the threaded tree is `A,B,C,D,E,F,G,H,I`, the predecessor of `E` is `D`, the successor of `E` is `F`.

## Example

Let's make the Threaded Binary tree out of a normal binary tree:

The in-order traversal for the above tree is — D B A E C. So, the respective Threaded Binary tree will be --

In an m-way threaded binary tree with n nodes, there are n*m - (n-1) void links.

## Iterative traversal

There is an algorithm for traversing a threaded binary tree without using recursion. Any such method must instead use iteration; that is, all steps must be done in a loop in order to visit all the nodes in the tree.

In-order traversal by definition first visits each node's left sub-tree (if it exists and if it is not already visited); then it visits the node itself, and finally the node's right sub-tree (if it exists). If the right sub-tree does not exist, but there is a threaded link, we make the threaded node the current node in consideration. An example is shown below.

### Algorithm

1. If the current node has a left child which is not in the visited list, then go to step 2, else step 3.
2. Put that left child in the list of visited nodes and make it the current node. Go to step 6.
3. Visit the node. If the node has a right child, then go to step 4, else go to step 5.
4. Make the right child the current node. Go to step 6.
5. If there is a thread node, then make it the current node.
6. If all nodes have been printed then end, else go to step 1.
 Step Li 1 'A' has a left child B, which has not been visited. So, we put B in our list of visited nodes and B becomes our current node. B 2 'B' has a left child, 'D', which is not in our list of visited nodes. So, we put 'D' in that list and make it our current node. B D 3 'D' has no left child, so we visit 'D'. Then we check for its right child. 'D' has no right child and thus we check for its thread-link. It has a thread going till node 'B'. So, we make 'B' the current node. B D D 4 'B' has a left child, but its already in the list of visited nodes. So, visit 'B'. Now check if it has a right child. It does not. So, make its threaded node (i.e. 'A') the current node. B D D B 5 'A' has a left child, 'B', but it is already visited. So, visit 'A'. Now 'A' has a right child, 'C' and it is not visited. So add it to the list and make it the current node. B D C D B A 6 'C' has an left child 'E' which is not visited. Add it to the list and make it the current node. B D C E D B A 7 and finally..... D B A E C

## Related Research Articles In computer science, an AVL tree is a self-balancing binary search tree. It was the first such data structure to be invented. 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. Lookup, insertion, and deletion all take O(log n) time in both the average and worst cases, where is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations. In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree whose internal nodes each store a key greater than all the keys in the node's left subtree and less than those in its right subtree. A binary tree is a type of data structure for storing data such as numbers in an organized way. Binary search trees allow binary search for fast lookup, addition and removal of data items, and can be used to implement dynamic sets and lookup tables. The order of nodes in a BST means that each comparison skips about half of the remaining tree, so the whole lookup takes time proportional to the binary logarithm of the number of items stored in the tree. This is much better than the linear time required to find items by key in an (unsorted) array, but slower than the corresponding operations on hash tables. Several variants of the binary search tree have been studied. In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the left child and the right child. A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple, where L and R are binary trees or the empty set and S is a singleton set containing the root. Some authors allow the binary tree to be the empty set as well.

In computer science, a B-tree is a self-balancing tree data structure that maintains sorted data and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree generalizes the binary search tree, allowing for nodes with more than two children. Unlike other self-balancing binary search trees, the B-tree is well suited for storage systems that read and write relatively large blocks of data, such as disks. It is commonly used in databases and file systems.

In computer science, a red–black tree is a kind of self-balancing binary search tree. Each node stores an extra bit representing "color", used to ensure that the tree remains balanced during insertions and deletions.

A splay tree is a binary search tree with the additional property that recently accessed elements are quick to access again. Like self-balancing binary search trees, a splay tree performs basic operations such as insertion, look-up and removal in O(log n) amortized time. For many sequences of non-random operations, splay trees perform better than other search trees, even performing better than O(log n) for sufficiently non-random patterns, all without requiring advance knowledge of the pattern. The splay tree was invented by Daniel Sleator and Robert Tarjan in 1985. In computer science, a tree is a widely used abstract data type that simulates a hierarchical tree structure, with a root value and subtrees of children with a parent node, represented as a set of linked nodes. In discrete mathematics, tree rotation is an operation on a binary tree that changes the structure without interfering with the order of the elements. A tree rotation moves one node up in the tree and one node down. It is used to change the shape of the tree, and in particular to decrease its height by moving smaller subtrees down and larger subtrees up, resulting in improved performance of many tree operations.

In computer science, tree traversal is a form of graph traversal and refers to the process of visiting each node in a tree data structure, exactly once. Such traversals are classified by the order in which the nodes are visited. The following algorithms are described for a binary tree, but they may be generalized to other trees as well.

Adaptive Huffman coding is an adaptive coding technique based on Huffman coding. It permits building the code as the symbols are being transmitted, having no initial knowledge of source distribution, that allows one-pass encoding and adaptation to changing conditions in data.

In computer science, corecursion is a type of operation that is dual to recursion. Whereas recursion works analytically, starting on data further from a base case and breaking it down into smaller data and repeating until one reaches a base case, corecursion works synthetically, starting from a base case and building it up, iteratively producing data further removed from a base case. Put simply, corecursive algorithms use the data that they themselves produce, bit by bit, as they become available, and needed, to produce further bits of data. A similar but distinct concept is generative recursion which may lack a definite "direction" inherent in corecursion and recursion. In computer science, a radix tree is a data structure that represents a space-optimized trie in which each node that is the only child is merged with its parent. The result is that the number of children of every internal node is at most the radix r of the radix tree, where r is a positive integer and a power x of 2, having x ≥ 1. Unlike regular trees, edges can be labeled with sequences of elements as well as single elements. This makes radix trees much more efficient for small sets and for sets of strings that share long prefixes.

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In computer science, a leftist tree or leftist heap is a priority queue implemented with a variant of a binary heap. Every node x has an s-value which is the distance to the nearest leaf in subtree rooted at x. In contrast to a binary heap, a leftist tree attempts to be very unbalanced. In addition to the heap property, leftist trees are maintained so the right descendant of each node has the lower s-value. In computer science, recursion is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem. Such problems can generally be solved by iteration, but this needs to identify and index the smaller instances at programming time. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.

The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions. A tree sort is a sort algorithm that builds a binary search tree from the elements to be sorted, and then traverses the tree (in-order) so that the elements come out in sorted order. Its typical use is sorting elements online: after each insertion, the set of elements seen so far is available in sorted order. In computer science, a Cartesian tree is a binary tree derived from a sequence of numbers; it can be uniquely defined from the properties that it is heap-ordered and that a symmetric (in-order) traversal of the tree returns the original sequence. Introduced by Vuillemin (1980) in the context of geometric range searching data structures, Cartesian trees have also been used in the definition of the treap and randomized binary search tree data structures for binary search problems. The Cartesian tree for a sequence may be constructed in linear time using a stack-based algorithm for finding all nearest smaller values in a sequence.

In computer science, an x-fast trie is a data structure for storing integers from a bounded domain. It supports exact and predecessor or successor queries in time O(log log M), using O(n log M) space, where n is the number of stored values and M is the maximum value in the domain. The structure was proposed by Dan Willard in 1982, along with the more complicated y-fast trie, as a way to improve the space usage of van Emde Boas trees, while retaining the O(log log M) query time.

In computer science, an optimal binary search tree , sometimes called a weight-balanced binary tree, is a binary search tree which provides the smallest possible search time for a given sequence of accesses. Optimal BSTs are generally divided into two types: static and dynamic.

1. Van Wyk, Christopher J. Data Structures and C Programs, Addison-Wesley, 1988, p. 175. ISBN   978-0-201-16116-8.
2. Knuth, D.E. (1968). Fundamental Algorithms. The Art of Computer Programming. 1 (1st ed.). Reading/MA: Addison Wesley.
3. Morris, Joseph H. (1979). "Traversing binary trees simply and cheaply". Information Processing Letters . 9 (5). doi:10.1016/0020-0190(79)90068-1.
4. Mateti, Prabhaker; Manghirmalani, Ravi (1988). "Morris' tree traversal algorithm reconsidered". Science of Computer Programming. 11: 29–43. doi:10.1016/0167-6423(88)90063-9.
5. Knuth, D.E. (1969). Fundamental Algorithms. The Art of Computer Programming. 1 (2 ed.). Addison Wesley. Hre: Sect.2.3.1 "Traversing Binary Trees".
6. Perlis, Alan Jay; Thornton, C. (Apr 1960). "Symbol manipulation by threaded lists". Communications of the ACM. 3 (4): 195–204. doi:10.1145/367177.367202.