# Binary search tree algorithm wikipedia

The definition and description use the term "value", and some of the binary search tree algorithm wikipedia examples use the term "key". I believe they are interchangeable in this article. We should pick one and be consistent.

I propose we pick "key". I think it is more precise. Does anyone know what a tree with duplicate sort values looks like? Shouldn't the definition of BST be that every node's right subtree has values greater than or equal to the node's value? The current definition says it's strictly greater, which isn't the case. Here is an excerpt from the lecture I once took at UC Berkeley: The full lecture can be found here http: I've also verified this with my reference book on algorithms.

By this definition self-balancing binary search trees are not binary search tree. Consider an empty AVL tree and insert three equal values.

After the rotation there are equal values to both sides of the root node. So you either need to exclude equal values from binary search trees or allow equal values to both sides of a node.

The right subtree of a node contains only values greater than or equal to the node's value. It is confusing that the article starts by claiming that equal nodes should always be in the right subtree, while the example image puts equal nodes such as the seven, which equals the root in the left subtree. Why does the blurb about Optimal BSTs have a reference to "section Is this text copied and pasted out of a book?

Is this a c violation? Yes - thank you very much. I see a few other minor flaws in the algorithm on the page. If you are deleting a leaf node, you must remove the link from the parent of that node to the node itself. I also think that the delete method as it is implemented would cause an error, because you are deleting the node and then trying to get data from it Slightly off topic, but could the python examples be made clearer by the use of exceptions instead of returning numbers eg.

C is the appropriate choice here. I can't understand the "Traversal" example because I don't know binary search tree algorithm wikipedia "callback" is or does. An example in another language is needed. This seems like a discussion without proper arguments.

People prefer different languages. I find Python more readable. I don't think the article binary search tree algorithm wikipedia have any examples in concrete programming languages. Pseudocode should be more than enough, this is an encyclopedic article, not a coding cookbook.

I think we should try making it as easy to understand for non-computer-scientists as reasonably possible. Blocks of code don't really help with this objective. It seems to me this article details the same concept as Binary search algorithm. I suggest a merger.

He Who Is The deletion algorithm should check for the case of both left and right being NULL before checking either side.

No, the code works fine - I tested it. I worked on understanding the code in the article and found some issues. I described them on my blog, see http: I hate to keep just pasting code around like this, but here's my take on the C version of removeNode. Tiger of Doom talk I implemented a recursive delete function - since BSTs are recursive abstractions I think that the deletion function in the article should use recursion.

We can then use a dynamic programming solution, detailed in section Cormen Sec Edition, to construct the tree with the binary search tree algorithm wikipedia possible expected search cost. I wish that someone who has the book or has a nearby library from which he can borrow it could take the time to write about binary search tree algorithm wikipedia.

Since both the successor and the predecessor must binary search tree algorithm wikipedia fewer than two children, either one can be deleted using the previous two cases. In a good implementation, it is generally recommended[citation needed] to avoid consistently using one of these nodes, because this can unbalance the tree. I disagree with this. Depending on which side of a node the equal values go to left or rightyou can only base this deletion schema in predecessor OR sucessor not arbitrary.

There are cases where taking the wrong choice will make equal values appear to the side they shouldn't be in. I'm just wondering whether the example code for Deletion is correct The example seems to fail when the deleted key is not found in the tree.

It also acts differently than the upper image the example deletes the minimum of binary search tree algorithm wikipedia right side, where the image deletes the max of the left side. I'm a student at Cal Poly State University San Luis Obispo, binary search tree algorithm wikipedia as part of a class on teaching technical subjects we were asked to find an article that was unclear or otherwise inaccessible for the wider audience and clarify it, with an emphasis on improving understanding from a teaching point of view.

Feel free to reintegrate that information if you'd like, but in my opinion it doesn't belong in this article, or if it does it belongs in it's own section. My wiki markup skills are not very good, so feel free to correct any incorrect wiki-links I may inadvertently create. I've done my best to make them at least link somewhere correct, but I can't get the hang binary search tree algorithm wikipedia linking to sections in other articles. So the image can contain one or more duplicated values, to show where they go.

I am asking senior wikipedia editors to approve this link and add it to the current article. I also made the intro clearer. I am pretty binary search tree algorithm wikipedia the code is correct, but any tips would be helpful. What if either value is NaN? Hi, What do you think of adding an observations section with regard to the data structure? Another observation is how to get the next or previous.

What about modified structures where the root contains the 'count', or has 'max' and 'min' fields, so it can have fewer operations in such cases? So should I edit away? I don't have much background in CS, but I wrote a binary-tree vocabulary once. The Python code assumes a parent pointer. The definition of a BST does not include a parent pointer. It seems that there is disagreement in the definition of a binary search tree. The article currently says that there must be no duplicates and there are sources online that back it upbut many standard implementations of BSTs support binary search tree algorithm wikipedia, and Nick Parlante, at Stanford, says that they can contain duplicates http: If these rules are broken then the invariant associated with the bst breaks.

You can effectively binary search tree algorithm wikipedia of a bst as a concrete implementation of the abstract data type Set. The code checks the options trading and instruction of BST is, in my opinion, wrong.

At least, clarification should be added. Second, it seems like these fields should be vice versa i. This sentence, and the paragraph it appears in, suffer from a rather unfortunate choice of concepts to explain orders and comparisons. Making a distinction between less-than and comparison functions is language-specific; in languages with operator overloading, it can lead to circular definitions because less-than in fact has to be implemented in terms of what this paragraph calls a "comparator".

I've been wanting to write something about this, but couldn't get my sources together, so I'll dump my thoughts here for now in the hopes that others can help. It is possible to implement the insert, delete and lookup operations in terms of two helper routines:.

With these, it is also possible to implement union, intersection and set difference much more efficiently than by just running repeated insertions, lookups and deletions. The symbol TreeNode is used in the article twice in different sense.

The second case is a struct defined in Binary search tree Verification. There appears to be some differences in the introduction and the other sections of the article relating to balancing. From my reading the article seems that both balanced and unbalanced are claimed. I may have read it wrong. From the B-Tree article: From the Types section: It is unbalanced and, in the worst case, performance degrades to that of a linked list. I have just modified 2 external links on Binary search tree.

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In computer sciencea self-balancing or height-balanced binary search tree is any node -based binary search tree that automatically keeps its height binary search tree algorithm wikipedia number of levels below the root small in the face of arbitrary item insertions and deletions. These structures provide efficient implementations for mutable ordered listsand can be used for other abstract data structures such as associative arrayspriority queues and sets.

The red—black treewhich is a type of self-balancing binary search tree, was called symmetric binary B-tree [2] and was renamed but can still be confused with binary search tree algorithm wikipedia generic concept of self-balancing binary search tree because of the initials.

Most operations on a binary search tree Binary search tree algorithm wikipedia take time directly proportional to the height of the tree, so it is desirable to keep the height small.

It follows that for a tree with n nodes and height h:. However, the simplest algorithms for BST item insertion may yield a tree with height n in rather common situations. For example, when the items are inserted in sorted key order, the tree degenerates into a linked list with n nodes. The difference in performance between the two situations may be enormous: If the data items are known ahead of time, the height can be kept small, in the average sense, by adding values in a random order, resulting in a binary search tree algorithm wikipedia binary search tree.

However, there are many situations such as online algorithms where this randomization is not viable. Self-balancing binary trees solve this problem by performing transformations on the tree such as tree rotations at key insertion times, in order to keep the height proportional to log 2 n.

Although a certain overhead is involved, it may be justified in the long run by ensuring fast execution of later operations. In the asymptotic " Big-O " sense, a self-balancing BST structure containing n items allows the lookup, insertion, and removal of an item in O log n worst-case time, and ordered enumeration of all items in O n time.

For some implementations these are per-operation time bounds, while for others they are amortized bounds over a sequence of operations. These times are asymptotically optimal among all data structures that manipulate the key only through comparisons. Self-balancing binary search trees can be used in a natural way to construct and maintain ordered lists, such as priority queues. They can also be used for associative arrays ; key-value pairs are simply inserted with an ordering based on the key alone.

In this capacity, self-balancing BSTs have a number of advantages and disadvantages over their main competitor, hash tables. One advantage of self-balancing BSTs is that they allow fast indeed, asymptotically optimal enumeration binary search tree algorithm wikipedia the items in key orderwhich hash tables do not provide.

One disadvantage is that their lookup algorithms get more complicated when there may be multiple items with the same key. Self-balancing BSTs have binary search tree algorithm wikipedia worst-case lookup performance than hash tables O log n compared to O nbut have worse average-case performance O log n compared to O 1. Self-balancing BSTs can be used to implement any algorithm that requires mutable ordered lists, to achieve optimal worst-case asymptotic performance.

For example, if binary tree sort is implemented with a self-balanced BST, we have a very simple-to-describe yet asymptotically optimal O n log n sorting algorithm. Similarly, many algorithms in computational geometry exploit variations on self-balancing BSTs to solve problems such as the line segment intersection problem and the point location problem efficiently. For average-case performance, however, self-balanced BSTs may be less efficient than other solutions.

Binary tree sort, in particular, is likely to be slower than merge sortquicksortor heapsortbecause of the tree-balancing overhead as well as cache access patterns. Self-balancing BSTs are flexible data structures, in that it's easy to extend them to efficiently record additional information or perform new operations. For example, one can record the number of nodes binary search tree algorithm wikipedia each subtree having a certain property, allowing one to count the number of nodes in a certain key range with that property in Binary search tree algorithm wikipedia log n time.

Wikimedia Commons has media related to Balanced Trees.

In computer sciencebinary search trees BSTsometimes called ordered or sorted binary treesare a particular type of container: They allow fast lookup, addition and removal of items, and can be used to implement either dynamic sets of items, or lookup tables that allow finding an item by its key e.

Binary search trees keep their keys in sorted order, so that lookup and other operations can use the principle of binary search: On average, this means that each comparison allows the operations to skip about half of the tree, so that each lookup, insertion or deletion takes time proportional to the 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; this article deals primarily with the basic type, making references to more advanced types when appropriate. A binary search tree is a rooted binary treewhose internal nodes each store a key and optionally, an associated value and each have two distinguished sub-trees, commonly denoted left and right. The tree additionally satisfies the binary search property, which states that the key in each node must be greater than or equal to any key stored in the left sub-tree, and less than or equal to any key stored in the right sub-tree.

Frequently, the information represented by each node is a record rather than a single data element. However, for sequencing purposes, nodes are compared according to their keys rather than any part of their associated records. The major advantage of binary search trees over other data structures is that the related sorting algorithms and search algorithms such as in-order traversal can be very efficient; they are also easy to code.

Binary search trees are a fundamental binary search tree algorithm wikipedia structure used to construct more abstract data structures such as setsmultisetsand associative arrays. Binary search requires an order relation by which binary search tree algorithm wikipedia element item can be compared with every other element in the sense of a total preorder.

The part of the element which effectively takes place in the comparison is called its key. In the context of binary search trees a total preorder is realized most flexibly by means of a three-way comparison subroutine. Binary search trees support binary search tree algorithm wikipedia main operations: Searching a binary search tree for a specific key can be programmed recursively or iteratively. We begin by examining the root node.

If the tree is nullthe key we are searching for does not exist in the tree. Otherwise, if the key equals that of the root, the search is successful binary search tree algorithm wikipedia we return the node. If the key is less than that of the root, we search the left subtree. Similarly, if the key is greater than that of the root, we search the right subtree. This process is repeated until the key is found or the remaining subtree is null.

If the searched key is not found after a null subtree is reached, then the key is not present in the tree. This is easily expressed as a recursive algorithm implemented in Python:. If the order relation is only a total preorder a reasonable extension of the functionality is the following: A binary tree sort equipped with such a comparison function becomes stable. Because in the worst case this algorithm must search from the root of the tree to the leaf farthest from the root, the search operation takes time proportional to the tree's height see tree terminology.

On average, binary search trees with n nodes have O log n height. Insertion begins as a search would begin; if the key is not equal to that of the root, binary search tree algorithm wikipedia search the left or right subtrees as before.

Eventually, we will reach an external node and add the new key-value pair here encoded as a record 'newNode' as its right or left child, depending on the node's key.

In other words, we examine the root and recursively insert the new node to the left subtree if its key is less than that of the root, or the right subtree if its key is greater than or equal to the root. The above destructive procedural variant binary search tree algorithm wikipedia the tree in place. It uses only constant heap space and the iterative version uses constant stack space as wellbut the prior version of the tree is lost. Alternatively, as in the following Python example, we can reconstruct all ancestors of the inserted node; any reference to the original tree root remains valid, making the tree a persistent data structure:.

The part that is rebuilt uses O log n space in the average case and O n in the worst case. In either version, this operation requires time proportional to the height of the tree in the worst case, which is O log n time in the average case over all trees, but O n time in the worst case.

Another way to explain insertion is that in order to insert a new node in the tree, its key is first compared with that of the root. If its key is less than the root's, it is then compared with the key of the root's left child. If its key is greater, it is compared with the root's binary search tree algorithm wikipedia child.

This process continues, until the new node is compared with a leaf node, and then it is added as this node's right or left binary search tree algorithm wikipedia, depending on its key: There are other ways of inserting nodes into a binary tree, but this is the only way of inserting nodes at the leaves and at the same time preserving the BST structure.

When removing a node from a binary search tree it is mandatory to maintain the in-order sequence of the nodes. There are many possibilities to do this. However, the following method which has been proposed by T.

Hibbard in [2] guarantees that the heights of the subject subtrees are changed by at most one. There are three possible cases to consider:. Broadly speaking, nodes with children are harder to delete. As with all binary trees, a node's in-order successor is its right subtree's left-most child, and a node's in-order predecessor is the left subtree's right-most child. In either case, this node will have only one or no binary search tree algorithm wikipedia at all.

Delete it according to one of the two simpler cases above. Consistently using the in-order successor or the in-order predecessor for every instance of the two-child case can lead to an unbalanced tree, so some implementations select one or the other at different times. Although this operation does not always traverse the tree down to a leaf, this is always a possibility; binary search tree algorithm wikipedia in the worst case it requires time proportional to the height of the tree.

It does not require more even when the node has two children, since it still follows a single path and does not visit any node twice. Once the binary search tree has been created, its elements can be retrieved in-order by recursively binary search tree algorithm wikipedia the left subtree of the root node, accessing the node itself, then recursively traversing the right subtree of the node, continuing this pattern with each node in the tree as it's recursively accessed.

As with all binary trees, one may conduct a pre-order traversal or a post-order traversalbut neither are likely to be useful for binary search trees. An in-order traversal of a binary search tree will always result in a sorted list of node items numbers, strings or other comparable items. The code for in-order traversal binary search tree algorithm wikipedia Python binary search tree algorithm wikipedia given below. It will call callback some function the programmer wishes to call on the node's value, such as printing to the screen for every node in the tree.

Traversal requires O n time, since it must visit every node. This algorithm is also O nso it is asymptotically optimal. Traversal can also be implemented iteratively. For certain applications, e. This is, of course, implemented without the callback construct and takes O 1 on average and O log n in the worst case.

Sometimes we already have a binary tree, and we need to determine whether it is a BST. This problem has a simple recursive solution. The BST property—every node on the right subtree has to be larger than the current node binary search tree algorithm wikipedia every node on the left subtree has to be smaller than the current node—is the key to figuring out whether a tree is a BST or not. The greedy algorithm —simply traverse the tree, at every node check whether the node contains a value larger than the value at the left child and smaller than the value on the right child—does not work for all cases.

Consider the binary search tree algorithm wikipedia tree:. In the tree above, each node meets the condition that the node contains a value larger than its left child and smaller than its right child hold, and binary search tree algorithm wikipedia it is not a BST: Instead of making a decision based solely on the values of a node and its children, we also need information flowing down from the parent as well.

In the case of the tree above, if we could remember about the node containing the value 20, we would see that the node with value 5 is violating the BST property contract. As pointed binary search tree algorithm wikipedia in section Traversalan in-order traversal of a binary search tree returns the nodes sorted.

A binary search binary search tree algorithm wikipedia can be used to implement a simple sorting algorithm. Similar to heapsortwe insert all the values we wish to sort into a new ordered data structure—in this case a binary search tree—and then traverse it in order. There are several schemes for overcoming this flaw with simple binary trees; the most common is the self-balancing binary search tree.

If this same procedure is done using such a tree, the overall worst-case time is O n log nwhich is asymptotically optimal for a comparison sort. In practice, the added overhead in binary search tree algorithm wikipedia and space for a tree-based sort particularly for node allocation make it inferior to other asymptotically optimal sorts such as heapsort for static list sorting. On the other hand, it is one of binary search tree algorithm wikipedia most efficient methods of incremental sortingadding items to a list over time while keeping the list sorted at all times.

Binary search trees can serve as priority queues: Insertion works as previously explained. Find-min walks the tree, following left pointers as far as it can without hitting a leaf:.

Delete-min max can simply look up the minimum maximumthen delete it. This way, insertion and deletion both take logarithmic time, just as they do in a binary heapbut unlike a binary heap and most other priority queue implementations, a single tree can support all of find-minfind-maxdelete-min and delete-max at the same time, making binary search trees suitable as double-ended priority queues. There are many types of binary search trees.

AVL trees and red-black trees are both forms of self-balancing binary search trees. A splay tree is a binary search tree algorithm wikipedia search tree that automatically moves frequently accessed elements nearer to the root.

In a treap tree heapeach node also holds a randomly chosen binary search tree algorithm wikipedia and the parent node has higher priority than its children. Tango trees are trees optimized for fast searches.

T-trees are binary search trees optimized to reduce storage space overhead, widely used for in-memory databases. A degenerate tree is a tree where for each parent node, there is only one associated child node. It is unbalanced and, in the worst case, performance degrades to that of a linked list.

If your add node function does not handle re-balancing, then you can easily construct a degenerate tree by feeding it with data that is already sorted. What this means is that in a performance measurement, the tree will essentially behave like a linked list data structure.

Heger [4] presented a performance comparison of binary search trees. Treap was found to have the best average performance, while red-black tree was found to have the smallest amount of performance variations. If we do not plan on modifying a search tree, and we know exactly how often each item will be accessed, we can construct [5] an optimal binary search treewhich is a search tree where the average cost of looking up an item the expected search cost is minimized.

Even if we only have estimates of the search costs, such a system can considerably speed up lookups on average. For example, if you have a BST of English words used in a spell checkeryou might balance the tree based on word frequency in text corporaplacing words like the near the root and words like agerasia near the leaves.