A binary search tree of size 9 and depth 3, with 8 at the root. The leaves are not drawn. In, binary search trees ( BST), sometimes called ordered or sorted binary trees, are a particular type of: that store 'items' (such as numbers, names etc.) in. They allow fast lookup, addition and removal of items, and can be used to implement either of items, or that allow finding an item by its key (e.g., finding the phone number of a person by name). Binary search trees keep their keys in sorted order, so that lookup and other operations can use the principle of: when looking for a key in a tree (or a place to insert a new key), they traverse the tree from root to leaf, making comparisons to keys stored in the nodes of the tree and deciding, on the basis of the comparison, to continue searching in the left or right subtrees.

This articles describes the algorithm to insert and delete elements in a Binary Search Tree (BST) and it's implementation in C#.

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 the of the number of items stored in the tree. This is much better than the required to find items by key in an (unsorted) array, but slower than the corresponding operations on. 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. Contents • • • • • • • • • • • • • • • • • • • Definition [ ] A binary search tree is a, whose 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 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.: 287 (The leaves (final nodes) of the tree contain no key and have no structure to distinguish them from one another. Leaves are commonly represented by a special leaf or nil symbol, a NULL pointer, etc.) 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 and such as can be very efficient; they are also easy to code.

Binary search trees are a fundamental data structure used to construct more abstract data structures such as,, and. • When inserting or searching for an element in a binary search tree, the key of each visited node has to be compared with the key of the element to be inserted or found.

• The shape of the binary search tree depends entirely on the order of insertions and deletions, and can become degenerate. • After a long intermixed sequence of random insertion and deletion, the expected height of the tree approaches square root of the number of keys, √ n, which grows much faster than log n. • There has been a lot of research to prevent degeneration of the tree resulting in worst case time complexity of log n (for details see section ). Order relation [ ] Binary search requires an order relation by which every element (item) can be compared with every other element in the sense of a. The part of the element which effectively takes place in the comparison is called its key. Whether duplicates, i.e.

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Different elements with same key, shall be allowed in the tree or not, does not depend on the order relation, but on the application only. In the context of binary search trees a total preorder is realized most flexibly by means of a. Operations [ ] Binary search trees support three main operations: insertion of elements, deletion of elements, and lookup (checking whether a key is present). Searching [ ] Searching a binary search tree for a specific key can be programmed.

We begin by examining the. If the tree is null, the key we are searching for does not exist in the tree.

Otherwise, if the key equals that of the root, the search is successful and 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 ). 1 def search_iteratively ( key, node ): 2 current_node = node 3 while current_node is not None: 4 if key == current_node. Key: 5 return current_node 6 if key current_node.key: 9 current_node = current_node.