AVL TREE














































AVL TREE



AVL tree is a height-balanced binary
search tree. That means, an AVL tree is also a binary search tree but it is a
balanced tree. A binary tree is said to be balanced if, the difference between
the heights of left and right subtrees of every node in the tree is either  -1, 0 or +1. In other words, a binary tree is
said to be balanced if the height of left and right children of every node
differ by either  -1, 0 or +1. In an AVL
tree, every node maintains an extra information known as balance factor. The
AVL tree was introduced in the year 1962 by G.M. Adelson-Velsky and E.M.
Landis.


An AVL tree is defined as follows:



An AVL tree is a balanced binary search tree. In an AVL
tree, balance factor of every node is either  -1, 0 or +1.



Balance factor of a node is the
difference between the heights of the left and right subtrees of that node. The
balance factor of a node is calculated either height of left subtree - height
of right subtree (OR) height of right subtree - height of left subtree. In the
following explanation, we calculate as follows:



Balance factor = heightOfLeftSubtree - heightOfRightSubtree.



Example:-


AVL Tree Rotations



In AVL tree, after performing
operations like insertion and deletion we need to check the balance factor of
every node in the tree. If every node satisfies the balance factor condition
then we conclude the operation otherwise we must make it balanced. Whenever the
tree becomes imbalanced due to any operation we use rotation operations to make
the tree balanced.



Rotation operations are used to make
the tree balanced.



Rotation is the process of moving nodes either to left or to
right to make the tree balanced.

Operations on an AVL Tree



The following operations are performed
on AVL tree...



1. Search



2. Insertion



3. Deletion



Search Operation in AVL Tree



In an AVL tree, the search operation is
performed with O(log n) time complexity. The search operation in the AVL tree
is similar to the search operation in a Binary search tree. We use the
following steps to search an element in AVL tree...



 Step
1 - Read the search element from the user.



 Step
2 - Compare the search element with the value of root node in the tree.



 Step
3 - If both are matched, then display "Given node is found!!!" and                                                                                 terminate
the function



 Step 4 - If both are not matched, then check
whether search element is smaller or larger than that node value.



 Step
5 - If search element is smaller, then continue the search process in left
subtree.



 Step
6 - If search element is larger, then continue the search process in right
subtree.



Step 7 - Repeat the same until we find
the exact element or until the search element is compared with the leaf node.



Step 8 - If we reach to the node having
the value equal to the search value, then display "Element is found"
and terminate the function.



Step 9 - If we reach to the leaf node
and if it is also not matched with the search element, then display
"Element is not found" and terminate the function.



Insertion Operation in AVL Tree



In an AVL tree, the insertion operation
is performed with O(log n) time complexity. In AVL Tree, a new node is always
inserted as a leaf node. The insertion operation is performed as follows...



 



 Step
1 - Insert the new element into the tree using Binary Search Tree insertion
logic.



Step 2 - After insertion, check the
Balance Factor of every node.



 Step 3 - If the Balance Factor of every node
is 0 or 1 or -1 then go for next operation.



Step 4 - If the Balance Factor of any
node is other than 0 or 1 or -1 then that tree is said to be imbalanced. In
this case, perform suitable Rotation to make it balanced and go for next
operation.



Deletion Operation in AVL Tree



The deletion operation in AVL Tree is
similar to deletion operation in BST. But after every deletion operation, we
need to check with the Balance Factor condition. If the tree is balanced after
deletion go for next operation otherwise perform suitable rotation to make the
tree Balanced.



 


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