Efficient search trees are data structures that allow efficient search, insertion, and deletion of elements. There are several types of efficient search trees, including binary search trees, AVL trees, red-black trees, B-trees, and more.
Binary search trees are a type of tree where each node has at most two children, and the left child is always smaller than the parent, while the right child is always larger. Binary search trees can be used to efficiently search for elements, insert new elements, and delete existing elements.
AVL trees are similar to binary search trees but are self-balancing, meaning that they automatically adjust their structure to maintain a balanced tree. This makes AVL trees more efficient for search, insertion, and deletion than regular binary search trees, especially when dealing with large amounts of data.
Red-black trees are another self-balancing binary search tree that is designed to be more efficient than AVL trees in certain situations, such as when inserting or deleting elements.
B-trees are a type of tree that can have multiple children per node, which makes them more efficient for storing large amounts of data. They are commonly used in databases and file systems.
Overall, the choice of an efficient search tree depends on the specific requirements of the problem at hand, including the size of the data, the frequency of operations, and the need for self-balancing.
2.1 Terminology: Balanced trees - AVL Trees, Red Black tree, splay tree, Lexical search tree -Trie
Balanced trees are a type of data structure where the height of the left and right subtrees of any node differ by at most one. Balanced trees help maintain efficient search, insertion, and deletion operations on a large dataset. Some of the commonly used balanced trees are AVL trees, Red Black trees, splay trees, and lexical search trees (Trie).
An AVL tree is a self-balancing binary search tree where the difference in height between the left and right subtrees of a node is at most one. If the balance factor of a node is greater than one or less than negative one, the tree is rebalanced using rotations.
A Red Black tree is another type of self-balancing binary search tree that uses color coding to maintain the balance of the tree. Each node is either red or black, and certain rules must be followed during insertion and deletion to maintain the color balance of the tree.
A splay tree is a self-adjusting binary search tree where frequently accessed nodes are moved closer to the root of the tree. The idea is that frequently accessed nodes are likely to be accessed again, and by moving them closer to the root, future access times will be reduced.
A Lexical search tree, also known as a Trie, is a tree-like data structure that is commonly used to store strings. Each node in the tree represents a prefix of one or more strings. The structure of the tree allows for efficient search and insertion of strings.
2.2 AVL Tree- concept and rotations
An AVL tree is a self-balancing binary search tree where the difference in height between the left and right subtrees of a node is at most one. If the balance factor of a node is greater than one or less than negative one, the tree is rebalanced using rotations.
The balance factor of a node is defined as the difference in height between the left and right subtrees of the node. If the balance factor of a node is greater than one, the node is considered unbalanced and requires rebalancing. AVL trees use rotations to balance the tree.
There are four types of rotations in an AVL tree: left rotation, right rotation, left-right rotation, and right-left rotation. A left rotation is performed when a node is unbalanced with a balance factor of 2 and its left subtree has a balance factor of 1 or 0. A right rotation is performed when a node is unbalanced with a balance factor of -2 and its right subtree has a balance factor of -1 or 0. Left-right and right-left rotations are performed when a node is unbalanced with a balance factor of 2 or -2 and its left or right subtree has a balance factor of -1 or 1.
The goal of the rotations is to restore balance to the tree while maintaining the binary search property. During a rotation, the child node becomes the parent of its former parent node, and the parent node becomes the child of the child node's subtree.
2.3 Red Black trees - concept, insertion and deletion
A Red Black tree is another type of self-balancing binary search tree that uses color coding to maintain the balance of the tree. Each node is either red or black, and certain rules must be followed during insertion and deletion to maintain the color balance of the tree.
The rules for a Red Black tree are as follows:
1. Every node is either red or black.
2. The root node is black.
3. Every leaf node (null node) is black.
4. If a node is red, then its children must be black.
5. Every path from a node to a descendant leaf node must contain the same number of black
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