Binary Search Tree A binary search tree or BST is a binary tree in symmetric order. A binary search tree can: • Be empty • Have a key and not more than two other subtrees, which are called left subtree and right subtree. A binary search tree is in symmetric order, it means: • Each node contains a key.

Deleting a node from a BST --- Part 1 (easy cases) Warning. Fact: Deleting. A node from a Binary Search Tree is very complicated because: The resulting tree after the deletion must also be a Binary Search Tree. Delete the node by short-circuiting the BST at the deletion node. C Program To Perform Insertion, Deletion and Traversal In B-Tree B- tree is a multiway search tree. A node in B-tree of order n can have at most n-1 values and n children.

• Each node’s key is smaller than all node’s keys in the right subtree and bigger than all node’s keys in the left subtree. A binary search tree is also known as sorted or ordered binary tree. Binary search tree operations There are some common operations on the binary search tree: • Insert – inserts a new node into the tree • Delete – removes an existing node from the tree • Traverse – traverse the tree in pre-order, in-order and post-order. For the binary search tree, only in-order traversal makes sense • Search – search for a given node’s key in the tree All binary search tree operations are O(H), where H is the depth of the tree. The minimum height of a binary search tree is H = log 2N, where N is the number of the tree’s nodes.

Therefore the complexity of a binary search tree operation in the best case is O(logN); and in the worst case, its complexity is O(N). The worst case happens when the binary search tree is unbalanced.

Many algorithms have been invented to keep a binary search tree balanced such as the height-balanced tree or of Adelson- Velskii and Landis, B-trees, and Splay trees. C binary search tree implementation We can use a to model the binary search tree node a follows. Typedef int ( * comparer ) ( int, int ); Later on, you may change the type of data to string, float, or even void*, and change the comparer callback accordingly.

Insert a new node into the binary search tree If the binary search tree is empty, we just create a new node, otherwise, we start from the root node: We find the proper position for the new node by comparing the new node’s key with the root node’s key. If the key of the new node less than the root node’s key, we go to the left subtree. If the key of the new node is greater than the root node’s key, we go to the right subtree. We do this step recursively until we find the correct position in the tree to insert the new node. The following illustrates the insert_node function. } Search for a specific key in the binary search tree To search for a specific key in the binary search tree, we start from the root node.

If the tree is empty, the key does not exist. Otherwise, if the root node’s key is equal to the key, the search is successful, we terminate the search. If the key is less than the root node’s key, we search the left subtree. Likewise, if the key is greater than the root node’s key, we search the right subtree. We repeat this step recursively until the key is found or subtree is NULL.

Iframe passing parameters. } Traverse the binary search tree We can traverse the binary search tree once it is created by traversing recursively to the left subtree of the root node, accessing the root node’s data, and then recursively traversing to the right subtree of the root node. This is called in-order traversal of a binary tree. Because of the rules of the nodes’ keys in the binary search tree, this in-order traversal always creates a sorted list of nodes.

The following is the in-order traversal function of the binary search tree.

Additionally, the RedBlack class contains several other methods that offer convenient functionality: • GetMinKey(): Returns the minimum key value. • GetMaxKey(): Returns the maximum key value. • GetMinValue(): Returns the object having the minimum key value. • GetMaxValue(): Returns the object having the maximum key value. • GetEnumerator(): Returns a RedBlackEnumerator used to iterate through the Treap. • Keys(): Returns a RedBlackEnumerator used to iterate through the keys.