# Bst insertion time complexity

The above properties of Binary Search Tree provides an ordering among keys so that the operations like search, minimum and maximum can be done fast. If there is no ordering, then we may have to compare every key to search for a given key. Now we compare the number to be searched or the element to be searched with the mid element of the search space or the median and if the record being searched is lesser we go searching in the left half else we go searching in the right half, in case of equality we have found the element.

Search operation in binary search tree will be very similar. Searching an element in the binary search tree is basically this traversal in which at each step we will go either towards left or right and hence in at each step we discard one of the sub-trees.

Start from the root. Compare the searching element with root, if less than root, then recurse for left, else recurse for right. If the element to search is found anywhere, return true, else return false. We start searching a key from the root until we hit a leaf node. Once a leaf node is found, the new node is added as a child of the leaf node.

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Compare the inserting element with root, if less than root, then recurse for left, else recurse for right. After reaching the end, just insert that node at left if less than current else right. Time Complexity: The worst-case time complexity of search and insert operations is O h where h is the height of the Binary Search Tree.

WORST CASE TIME COMPLEXITY OF AN ALGORITHM 1

In the worst case, we may have to travel from root to the deepest leaf node. The height of a skewed tree may become n and the time complexity of search and insert operation may become O n.

Attention reader! Writing code in comment? Please use ide. Related Articles. The left and right subtree each must also be a binary search tree. There must be no duplicate nodes. A utility function to search a given key in BST. Base Cases: root is null or key is present at root. Key is greater than root's key. Key is smaller than root's key. This code is contributed by Bhavya Jain. BST. BST int. Insert root, 50 .Note that the above implementation is not a binary search tree because there is no restriction in inserting elements to the tree.

To insert a Node iteratively in a BST tree, we will need to traverse the tree using two pointers. Removing an element from a BST is a little complex than searching and insertion since we must ensure that the BST property is conserved. To delete a node we need first search it. Then we need to determine if that node has children or not. Sirplease teach us how to implement BST in C. Your email address will not be published. I would love to connect with you personally. Table of Contents 1 Binary Search Tree 1.

The inorder successor can be obtained by finding the minimum value in right child of the node. Next Huffman Coding Algorithm. Shoot him queries. Follow Author. Comments Anushka says:. September 6, at pm. Binary search property states that all the left nodes in a binary search tree have less value than its root node, and all the right nodes in a binary search tree have greater value than its root node. The main point here is to find the correct place in BST in order to insert the new node. Also, we need to ensure after the insertion of the new node that the tree satisfies the binary search property. The pseudocode of the insertion process can be found in a quick guide to binary search trees.

Notice that the value of the new node is greater than the value of the highest level node in the existing binary search tree. When we insert a new node, we first check the value of the root node.

If the value of the new node is greater than the root node, we search the right subtree for the possible insertion position. Otherwise, we explore the left subtree for the insertion. The new node here searches the possible insertion position in the right subtree.

We compare the value of the new node with all the nodes of the current tree and finds its insertion position at the last level of the binary search tree. Here, we want to insert a node with a value of. First, we see the value of the root node. Again we compare the value of the new node with the value of each node in the existing tree. Finally, we insert the new node at the last level of the binary search tree. In both cases, we have to travel from the root to the deepest leaf node in order to find an index to insert the new node. If there are nodes in the binary search tree, we need comparisons to insert our new node.

Therefore, in such cases, the overall time complexity of the insertion process would be. In this case, the existing binary search tree is a balanced tree. The existing binary search tree is a balanced tree when each level has nodes, where is the level of the tree. Now we want to insert a node with value.

For any new node that we want to insert, would be the maximum number of comparisons required, which is the height of the binary search tree Here, L is the level of the existing binary search tree.

The height of the binary search tree is also equal towhere is the total number of the node in the binary search tree. Therefore in the best and average case, the time complexity of insertion operation in a binary search tree would be.

Hence we just need to perform one comparison in order to insert the new node. Therefore in the best case, the time complexity of insertion operation in a binary search tree would be.

We presented the time complexity analysis and demonstrated different time complexity cases with examples. Full Archive The high level overview of all the articles on the site. About Baeldung About Baeldung.

Inline Feedbacks.The main operations in binary tree are: search, insert and delete. We will see the worst case time complexity of these operations in binary trees. Binary Tree — In a binary tree, a node can have maximum two children. Consider the left skewed binary tree shown in Figure 1. Binary Search Tree BST — BST is a special type of binary tree in which left child of a node has value less than the parent and right child has value greater than parent.

Consider the left skewed BST shown in Figure 2. What is the worst case time complexity for search, insert and delete operations in a general Binary Search Tree? So, the correct option is A. Solution: As discussed, search operation in binary tree and BST have worst case time complexity of O n. However, AVL tree has worst case time complexity of O logn. So, the correct option is D. Attention reader! Writing code in comment? Please use ide. Related Articles. Searching: For searching element 2, we have to traverse all elements assuming we do breadth first traversal. Therefore, searching in binary tree has worst case complexity of O n. Insertion: For inserting element as left child of 2, we have to traverse all elements.

Therefore, insertion in binary tree has worst case complexity of O n. Deletion: For deletion of element 2, we have to traverse all elements to find 2 assuming we do breadth first traversal. Therefore, deletion in binary tree has worst case complexity of O n. Recommended Articles. Article Contributed By :. Current difficulty : Easy.

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Easy Normal Medium Hard Expert. Improved By :. Most popular in Analysis.There are, of course, non-binary trees. However, it is important to note that a binary tree is not a special case of a tree but is a different concept. For example, those trees:. We can consider them identical when defining them as ordinary trees but different when analyzed as binary trees.

In a binary search tree, each node is identified by a key, which is stored respecting the following property: Let be a node of a binary tree. If is a node in the left subtree of then. If is a node in the right subtree ofthen.

Each row or record in the database is made up of a series of distinct fields identified by a key. Let be the number of records in the database, each consisting of fields.

Suppose that the key is unique for each record. It is possible to store organized as a binary search tree based on the property mentioned above.

Elementary or primitive operations in the binary search trees are search, minimum, maximum, predecessor, successor, insert, and delete. Computational complexity depends on the concept of the height of the treewhich we can informally define as the number of levels of which the tree is composed. For example, the binary tree from the first figure has 5 levels including root.

Suppose we have a keyand we want to retrieve the associated fields of for. The problem is formulated as the identification of the node such that.

So, we move into the tree, starting from the root node, comparing our key with the keys of the nodes we visit. Note that each move involves the descent of a level in the tree. If the key is unique, the number of nodes visited during the search is at most equal toand the search can be done in time. Not all binary search trees are equally efficient when performing a primitive operation. The key to improving efficiency is given by the fact that computational complexity depends on and not on.

The way the elements are arranged in the binary tree affects its height. In general, we can state the problem of the optimal construction, such as the search for the arrangement of the nodes that leads to the tree with the minimum height.

The worst scenario is a database already sorted by key. In this case, if we build a binary tree through insertions of the records in the original order, we will get a tree that contains only left or right subtrees, depending on whether the order of the keys is respectively descending or ascending:.

In this case,and by the discussion of the previous paragraph, the realization of a primitive operation occurs in time. This case is equivalent to a linked list. If keys of are disordered, building a binary tree based on insert operations produces a structure with.

## Time Complexity of Searching in a Balanced Binary Search Tree

When the heights of the left and right subtree of any node differ by not more than 1, the tree is said to be balancedand the following result can be demonstrated:. The average height of a randomly constructed binary search tree with distinct keys is.

From previous results, we conclude that the search for a key and, in general, any primitive operation performed on a binary search tree, takes time in the worst case and in the average case. The construction of a tree based on the insertion of the records of therefore requires time in the worst case and in the average case. Binary search trees are used in many computational procedures. However, the basic theory illustrated in this tutorial is not without problems.

In real applications, binary search trees are not necessarily balanced.In computer sciencea red—black tree is a kind of self-balancing binary search tree. Each node stores an extra bit representing "color" "red" or "black"used to ensure that the tree remains approximately balanced during insertions and deletions.

When the tree is modified, the new tree is rearranged and "repainted" to restore the coloring properties that constrain how unbalanced the tree can become in the worst case. The properties are designed such that this rearranging and recoloring can be performed efficiently. The re-balancing is not perfect, but guarantees searching in O log n time, where n is the number of nodes of the tree.

The insertion and deletion operations, along with the tree rearrangement and recoloring, are also performed in O log n time. Tracking the color of each node requires only 1 bit of information per node because there are only two colors.

The tree does not contain any other data specific to its being a red—black tree so its memory footprint is almost identical to a classic uncolored binary search tree. In many cases, the additional bit of information can be stored at no additional memory cost. InRudolf Bayer  invented a data structure that was a special order-4 case of a B-tree.

These trees maintained all paths from root to leaf with the same number of nodes, creating perfectly balanced trees. However, they were not binary search trees. Bayer called them a "symmetric binary B-tree" in his paper and later they became popular as trees or just trees. Guibas and Robert Sedgewick derived the red-black tree from the symmetric binary B-tree.

InArne Andersson introduced the idea of a right leaning tree to simplify insert and delete operations. InChris Okasaki showed how to make the insert operation purely functional. Its balance function needed to take care of only 4 unbalanced cases and one default balanced case. The original algorithm used 8 unbalanced cases, but Cormen et al. Sedgewick originally allowed nodes whose two children are red, making his trees more like trees, but later this restriction was added, making new trees more like trees.

Sedgewick implemented the insert algorithm in just 33 lines, significantly shortening his original 46 lines of code. A red—black tree is a special type of binary treeused in computer science to organize pieces of comparable datasuch as text fragments or numbers. The leaf nodes of red—black trees do not contain data. These leaves need not be explicit in computer memory—a null child pointer like NIL in the figure "An example of a red—black tree" below can encode the fact that this child is a leaf.

However, in the description of this figure, the leaves are considered to be explicit nodes—a view which may simplify the description and the understanding of some algorithms for operating on red—black trees. Now, in order to save a marginal amount of execution time see therethese NIL-leaves may be implemented as sentinel nodes instead of null pointers.

On the other hand, in order to save main memory, a single sentinel node instead of many individuals may perform the role of all leaf nodes: all references pointers from internal nodes to leaf nodes then point to this unique sentinel node. Red—black trees, like all binary search treesallow efficient in-order traversal that is: in the order Left—Root—Right of their elements. The search-time results from the traversal from root to leaf, and therefore a balanced tree of n nodes, having the least possible tree height, results in O log n search time.

In addition to the requirements imposed on a binary search tree the following must be satisfied by a red—black tree: . The only constraint on the children of black nodes is 5. In particular, a black node like a leaf node may have a black parent; for example, every perfect binary tree that consists only of black nodes is a red-black tree.

The black depth of a node is defined as the number of black nodes from the root to that node i.

### Binary Search Tree Search and Insertion

The black height of a red—black tree is the number of black nodes in any path from the root to the leaves, which, by property 5, is constant alternatively, it could be defined as the black depth of any leaf node.

These constraints enforce a critical property of red—black trees: the path from the root to the farthest leaf is no more than twice as long as the path from the root to the nearest leaf. The result is that the tree is roughly height-balanced.We have already discussed the Tree data structure and a Binary Search Tree. The basic Binary Search tree property says that, if an element is greater then the root node, it will be present in the right subtree and if it is smaller then the root node then it will be present in the left subtree.

So, for insertion, we will use this basic property of BST. We will follow here logic similar to insertion i. Deletion is a little tricky is Binary Search Tree. One important thing to consider is that after a deletion operation any of the BST property should not be violated. Here we deleted node 5 and attached its child 8 to its parent node Here in the 1st tree, we replaced 5 with the rightmost child 4 of its left successor 2 and in the second tree, we replaced 5 with the leftmost child 7 of its left successor 8.

CPP Implementation of above Algorithm. Note, this worst-case time complexity and will occur when the tree is left or right-skewed. Save my name, email, and website in this browser for the next time I comment.

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