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CS301-Lec20 handout

# CS301-Lec20 handout - CS301 Data Structures Lecture No 20...

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CS301 – Data Structures Lecture No. 20 ___________________________________________________________________ Data Structures Lecture No. 20 Reading Material Data Structures and Algorithm Analysis in C++ Chapter. 4 4.4 Summary AVL Tree Insertion in AVL Tree Example (AVL Tree Building) We will continue the discussion on AVL tree in this lecture. Before going ahead, it will be better to recap things talked about in the previous lecture. We built a balanced search tree (BST) with sorted data. The numbers put in that tree were in increasing sorted order. The tree built in this way was like a linked list. It was witnessed that the use of the tree data structure can help make the process of searches faster. We have seen that in linked list or array, the searches are very time consuming. A loop is executed from start of the list up to the end. Due to this fact, we started using tree data structure. It was evident that in case, both the left and right sub-trees of a tree are almost equal, a tree of n nodes will have log 2 n levels. If we want to search an item in this tree, the required result can be achieved, whether the item is found or not, at the maximum in the log n comparisons. Suppose we have 100,000 items (number or names) and have built a balanced search tree of these items. In 20 (i.e. log 100000) comparisons, it will be possible to tell whether an item is there or not in these 100,000 items. AVL Tree In the year 1962, two Russian scientists, Adelson-Velskii and Landis, proposed the criteria to save the binary search tree (BST) from its degenerate form. This was an effort to propose the development of a balanced search tree by considering the height as a standard. This tree is known as AVL tree. The name AVL is an acronym of the names of these two scientists. An AVL tree is identical to a BST, barring one difference i.e. the height of the left and right sub-trees can differ by at most 1. Moreover, the height of an empty tree is defined to be (–1). Keeping in mind the idea of the level of a tree, we can understand that if the root of a tree is at level zero, its two children (subtrees) i.e. nodes will be at level 1. At level 2, there will be 4 nodes in case of a complete binary tree. Similarly at level 3, the number of nodes will be 8 and so on. As discussed earlier, in a complete binary tree, the number of nodes at any level k will be 2 k . We have also seen the level order traversal of a tree. The term height is identical to the level of a tree. Following is the figure of a tree in which level/height of nodes is shown. Page 1 of 11

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CS301 – Data Structures Lecture No. 20 ___________________________________________________________________ Here in the figure, the root node i.e. 5 is at the height zero. The next two nodes 2 and 8 are at height (or level) 1. Then the nodes 1, 4 and 7 are at height 2 i.e. two levels below the root. At the last, the single node 3 is at level (height) 3. Looking at the figure, we can say that the maximum height of the tree is 3. AVL states that a tree should be formed in such a form that the difference of the heights (maximum no of levels i.e. depth) of left and right sub-trees of a node should not be greater than 1. The
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CS301-Lec20 handout - CS301 Data Structures Lecture No 20...

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