# Tutorialspoint binary search

Unlike self-balancing binary search trees, it is optimized for systems that read and write large blocks of data. It is most commonly used in database and file systems. B-tree nodes have many more than two children. A B-tree node may contain more than just a single element.

The set formulation of the B-tree rules: Searching for a Target in a Set The psuedocode:. For example, suppose we want to add 18 to the tree:. The above result is an illegal B-tree. Our plan is to perform a loose addition first, and then fix the root's problem. Loose removal allows to leave a root that has one element too few.

There are four cases that we need to consider:. Transfer an extra element from subset[ i -1]. Combine subset[ i ] with subset[ i -1]. We may need to continue activating fixShortage until the B-tree rules are satisfied. Build a tree with the following ten values: Delete 10, 8 and 6 from the tree. Trees -- Time Analysis The implementation of a B-tree is efficient since the depth of the tree is kept small.

Worst-case times for tree operations: Remember that a heap is a complete BST, so each level must be full before proceeding to the next level. Number of nodes needed for a heap to reach depth d is: Suppose a B-tree has n elements and M is the maximum number of children a node can have.

What is the maximum depth the tree could have? What is the minimum depth the tree could have? Back To Lectures Notes. A B-tree is a tree data structure that keeps data sorted and allows searches, insertions, and deletions in logarithmic amortized time. Important properties of a B-tree: The root can have as few as one element or even no elements if it also has no children ; every other node has at least MINIMUM elements.

The elements of each B-tree node are stored in a partially filled array, sorted from the smallest element at index 0 to the largest element at the final used position of the array.

The number of subtrees below a nonleaf node is always one more than the number of elements in the node. Subtree 0, subtree 1, For any nonleaf node: Every leaf in a B-tree has the same depth. Remember that " Every child of a node is also the root of a smaller B-tree ". If there is no such index, then set i equal to dataCount, indicating that none of the elements is greater than or equal to the target. We can implement a private method: If there's no such location, then return value is dataCount.

The elements of each B-tree node are stored in a partially filled array, sorted from the smallest element at index 0 to the largest element at the final used position of the array. The number of subtrees below a nonleaf node is always one more than the number of elements in the node.

Subtree 0, subtree 1, For any nonleaf node: Every leaf in a B-tree has the same depth. Remember that " Every child of a node is also the root of a smaller B-tree ". If there is no such index, then set i equal to dataCount, indicating that none of the elements is greater than or equal to the target. We can implement a private method: If there's no such location, then return value is dataCount. Adding an Element to a B-Tree. It is easier to add a new element to a B-tree if we relax one of the B-tree rules.

For example, suppose we want to add 18 to the tree: This leaves one extra element, which is passed up to the parent. It is always the middle element of the split node that moves upward. The parent of the split node gains one additional child and one additional element. The children of the split node have been equally distributed between the two smaller nodes.

Fixing the Root with an Excess Element: Create a new root. Deal with one of these four possibilities: The entire B-tree is still valid based on the looseRemoval rule. Fixing Shortage in a Child: There are four cases that we need to consider: Transfer data[ i -1] down to the front of subset[ i ].

Transfer the final element of subset[ i -1]. If subset[ i -1] has children, transfer the final child of subset[ i -1] over to the front of subset[ i ]. Transfer data[ i -1] down to the end of subset[ i -1]. Transfer all the elements and children from subset[ i ] to the end of subset[ i -1].

Removing the Biggest Element from a B-Tree: A tree is a type of B-tree where every node with children internal node has either two children and one data element 2-nodes or three children and two data elements 3-node. Leaf nodes have no children and one or two data elements. Take a look at this tree animation. The implementation of a B-tree is efficient since the depth of the tree is kept small.

Adding an element to a binary search tree BST , a heap, or a B-tree. Removing an element from a BST, a heap, or a B-tree. Searching for a specified element in a BST or a B-tree.

A BST with n elements could have a depth as big as n