Heap types & Trees
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The document discusses different types of self-balancing binary search trees including AVL trees, B-trees, B+-trees, and heaps. AVL trees balance the height of left and right subtrees to maintain logarithmic search time. B-trees and B+-trees store data in leaf nodes and indexes in internal nodes, allowing efficient storage and access of large datasets. Heaps implement priority queues using a complete binary tree structure to efficiently retrieve maximum or minimum elements.
![Properties
All leaves are at the same level.
A B-Tree is defined by the term minimum degree ‘t’. The
value of t depends upon disk block size.
Every node except root must contain at least (ceiling)([t-1]/2)
keys. The root may contain minimum 1 key.
All nodes (including root) may contain at most t – 1 keys.
Number of children of a node is equal to the number of keys
in it plus 1.
All keys of a node are sorted in increasing order. The child
between two keys k1 and k2 contains all keys in the range
from k1 and k2.
B-Tree grows and shrinks from the root which is unlike Binary
Search Tree. Binary Search Trees grow downward and also
shrink from downward.
Like other balanced Binary Search Trees, time complexity to
search, insert and delete is O(log n).
Insertion of a Node in B-Tree happens only at Leaf Node](https://image.slidesharecdn.com/avlbbheap-220202054753/85/Heap-types-Trees-8-320.jpg)
Heap types & Trees
- 1. Unit- 3
- 2. AVL trees AVL trees are binary search trees in which the difference between the height of the left and right subtree is either -1, 0, or +1. AVL trees are also called a self-balancing binary search tree. These trees help to maintain the logarithmic search time. It is named after its inventors (AVL) Adelson, Velsky, and Landis. Balance Factor = (Height of Left Subtree - Height of Right Subtree) https://www.javatpoint.com/insertion-in-avl-tree
- 3. Rotations Left – Left Rotation Right – Right Rotation Right – Left Rotation Left – Right Rotation https://www.tutorialspoint.com/data_str uctures_algorithms/avl_tree_algorithm .htm
- 4. Insertion
- 5. Deletion(from right sub tree)
- 6. Advantages of AVL Trees The height of the AVL tree is always balanced. The height never grows beyond log N, where N is the total number of nodes in the tree. It gives better search time complexity when compared to simple Binary Search trees. AVL trees have self-balancing capabilities.
- 7. B Tree B Trees is similar to that in Binary search tree B-Tree is a self-balanced search tree in which every node contains multiple keys and has more than two children. Operations Insertion Deletion Search
- 8. Properties All leaves are at the same level. A B-Tree is defined by the term minimum degree ‘t’. The value of t depends upon disk block size. Every node except root must contain at least (ceiling)([t-1]/2) keys. The root may contain minimum 1 key. All nodes (including root) may contain at most t – 1 keys. Number of children of a node is equal to the number of keys in it plus 1. All keys of a node are sorted in increasing order. The child between two keys k1 and k2 contains all keys in the range from k1 and k2. B-Tree grows and shrinks from the root which is unlike Binary Search Tree. Binary Search Trees grow downward and also shrink from downward. Like other balanced Binary Search Trees, time complexity to search, insert and delete is O(log n). Insertion of a Node in B-Tree happens only at Leaf Node
- 10. B + Tree B+ tree is used to store the records very efficiently by storing the records in an indexed manner using the B+ tree indexed structure. Due to the multi-level indexing, the data accessing becomes faster and easier In the B+ tree, keys are the indexes stored in the internal nodes and records are stored in the leaf nodes.
- 11. Properties All leaves are at the same level. The root has at least two children. Each node except root can have a maximum of m children and at least m /2 children. Each node can contain a maximum of m - 1 keys and a minimum of ⌈m/2⌉ - 1 keys.
- 13. Heaps A Heap is a special Tree-based data structure in which the tree is a complete binary tree Types 1. Max heap 2. Min heap
- 15. Applications Heap Implemented priority queues are used in Graph algorithms like Prim’s Algorithm (minimum spanning tree) and Dijkstra’s algorithm(shortest path) Heap sort Huffman encoding(data compression)