Introduction to AVL tree

AVL tree is a balanced binary search tree where the difference between heights of two sub trees is maximum 1.

Why balanced tree

A binary tree is good data structure because search operation here is of the order of O(logn). But this is true if the tree is balanced - which means the left and right subtrees are almost equal in height. If not balanced, search operation will take longer.

In worst case, if the tree has only one branch, then search is of the order O(n). Look at this example.

Here all nodes have only right children. To search a value in this tree, we need upto 7 iterations, which is O(n).

So this tree is very very inefficient.

One way of making the tree efficient is to, balance the tree and make sure that height of two branches of each node are almost equal.

Height of a node

Height of a node is the distance between the node and its extreme child.

In the above a diagram, height of 37 is 3 and height of left child of 37 is 0 and right child of 37 is 2.

Balancing a node

A node is rotated, in such a way that order of the elements is retained, but the node becomes more balanced.

In the above example, if we rotate the node containing 38 and make 37 as left child of 38, the tree is still in correct order, but is better balanced.

So we have performed single left rotation on the node.

When a node is inserted to an AVL tree, it needs a either a single rotation or a double rotation. But if a node is deleted from AVL tree, then the nodes need to be rotated successively until root is reached.

Comments

Program to delete a node from linked list

How do you remove a node from a linked list? If you have to delete a node, first you need to search the node. Then you should find its previous node. Then you should link the previous node to the next node. If node containing 8 has to be deleted, then n1 should be pointing to n2. Looks quite simple. Isn't it? But there are at least two special cases you have to consider. Of course, when the node is not found. If the node is first node of the list viz head. If the node to be deleted is head node, then if you delete, the list would be lost. You should avoid that and make the second node as the head node. So it becomes mandatory that you return the changed head node from the function. Now let us have a look at the code. #include<stdio.h> #include<stdlib.h> struct node { int data; struct node * next; }; typedef struct node * NODEPTR; NODEPTR create_node ( int value) { NODEPTR temp = (NODEPTR) malloc( size...

Binary tree deletion

Do not get all scared and worried. It is not rocket science (or as I would like to call it - it is not regex). Remember deleting a node from linked list. When you deleted a node, you did the following 2 steps Free the memory of the node Link the previous node of the deleted node to the next node You will have to do these things in binary tree too. But the difficulty here is do you link the previous node - or parent node in tree terminology, to the left child of deleted node? Or to the right child? You can not link both children, as the parent already may have one child node. So before we solve this, let us categorize the deletion with the help of a diagram. Consider the cases The node to be deleted is a leaf node. That is, it does not have left or right child. In this case, link the parent to NULL in place of deleted node. If you want to delete 13 - which is a leaf node, you set 14->left to NULL and free memory of node 13. If you want to delete 7 - which is also a leaf node and...

Function to add a node to binary search tree

Binary search tree (BST) is an ordered tree where each node has smaller values on the left sub tree and larger values on the right subtree. This is similar to binary search algorithm. And another useful fact is if you traverse the node in in-order method, the values of the tree will be in ascending order. Binary Search Tree with root as 50 Now as you can see from the diagram on the left child and their children of root (which in this case is 50), values are 40, 20 and 45. All are less than 50. On the other hand, on the right sub tree 50, the values are 60, 55 and 78 - larger than 50. And this rule follows for each and every node, not just root. Now if we have to add a node - let us 43, where shall we add it, in such a way that the tree will still remain as BST. The new node must be a leaf node (i.e. the node with no child nodes) Let us traverse from root and find out where this new node fits. 43 is smaller than 50. Hence we must branch to left of 50. Now we get 40. 4...

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