Algorithm avl data in paper research structure tree

Tree is one of the most important data structure that is used for efficiently performing operations like insertion, deletion and searching of values. However, while working with large volume of data, construction of well balanced tree for sorting all data s not feasible. Thus only useful data is stored as a tree and actual volume of data being used constantly changes through insertion of new data and deletion of existing data. You will find in some cases where the NULL link to a binary tree to special links is called as threads and hence it is possible to perform traversals, insertions, deletions without using either stack or recursion. In this chapter you will learn about the Height balance tree which is also known as AVL tree. What is AVL Tree? AVL tree is a binary search tree in which the difference of heights of left and right sub trees of any node is less than or equal to one. The technique of balancing the height of binary trees was developed by Adelson, Velskii and Landi and hence given the short form as AVL tree or Balanced Binary Tree. An AVL tree can be defined as follows: Let T be a non empty binary tree with T L and T R as its left and right sub trees. Tree is height balanced if:

  • T L and T R are height balanced
  • h L – h R <= 1, where h L – h R are the heights of T L and T R
The Balance factor of a node in a binary tree can have value 1, -1, 0, depending on whether the height of its left sub tree is greater, or less than or equal to the height of the right sub tree. Advantages of AVL tree Since AVL trees are height balance trees, operations like insertion and deletion have low time complexity. Let us consider an example: If you have the following tree having keys 1, 2, 3, 4, 5, 6, 7 and then the binary tree will be like the second figure: In order to insert a node with a key Q in the binary tree, the algorithm requires 7 comparisons but if you insert the same key in AVL tree, from the above 1st figure you can see that the algorithm will require 3 comparisons. Representation of AVL Trees Struct AVLNode { int data; struct AVLNode *left, *right; int balfactor; }; Algorithm for different Operations on AVL For Insertion: Step 1: First, insert a new element into the tree using BST’s (Binary Search Tree) insertion logic. Step 2: After inserting elements you have to check the Balance Factor of each node. Step 3: When the Balance Factor of every node will be found as 0 or 1 or -1 then the algorithm will proceed for next operation. Step 4: When the balance factor of any node comes other than the above three values then tree is said to be imbalanced. Then perform the suitable Rotation to make it balanced and then algorithm will proceed for next operation. For Deletion: Step 1: Firstly, find those node where k is stored Step 2: Secondly delete those contents of node (Suppose the node is x) Step 3: Claim: Deleting a node in an AVL tree can be reduced to deleting a leaf. There are three possible cases:
  • When x has no children then, delete x
  • When x has one child, let x’ becomes the child of x.
  • Notice: x’ cannot have a child, since sub trees of T can differ in height by at most one :
    • then replace the contents of x with the contents of x’
    • then delete x’ (a leaf)
  • Step 4:  When x has two children,
    • then find x’s successor z (which has no left child)
    • then replace x’s contents with z’s contents, and
    • delete z
In all of the three cases, you will end up removing a leaf.

Algorithm avl data in paper research structure tree

algorithm avl data in paper research structure tree

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