1. Advanced Data Structure using C++

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LECTURE NOTES OF ADVANCED DATA STRUCTURE (MT-CSE 110) Similar to case 2 except want to keep left child and any of its children. Case 4: Delete a parent with 2 children. E.g. delete Q. Problem: Can’t guarantee that the parent of the node could point to both children, since the parent may have another child of its own. AVL Tree In an AVL, the difference between the right and left sub‐tree can never be more than 1, throughout the tree. A binary search tree (BST) is an AVL tree if and only if, for every node in the tree, the difference between maximum height of the right sub‐tree minus the maximum height of the left sub‐tree is less than 2. That is the difference can be ‐1, 0, or <strong>1.</strong> AVL Trees is An ordered tree (binary search tree) is used when we wish to store objects with (numerical) keys in a binary tree so that lookups can be done in order log 2N time, where N is the number of objects in the tree. But an ordered tree that is seriously ``unbalanced,'' that is, where paths from the root to the leaves have dramatically different lengths, will ruin the desired lookup behavior. The worst‐case example of an unbalanced ordered tree is the tree built by inserting a sorted sequence of objects (we show the numerical keys only; the objects attached with the keys are unimportant): 1 2 3 4 The tree looks like this: 1 / \ . 2 Prepared By : Er. Harvinder Singh Assist Prof., CSE, H.C.T.M (Kaithal) Page ‐ 200 ‐
LECTURE NOTES OF ADVANCED DATA STRUCTURE (MT-CSE 110) / \ . 3 / \ . 4 / \ . . Obviously, a lookup in this tree is just a linear search, which is slower than log‐ time. How can we maintain an ordered tree so that, regardless of the order of insertions, the tree remains balanced? There are several sophisticated technques for doing so; here we consider one of the most elegant, AVL trees. Definition of an AVL tree (For example, if the tree held 2048 An AVL‐tree is an ordered tree that has the height-balanced property. Here are the basic definitions: The height of a tree is the length of the longest path from the tree's root to one of its leaves. A Node is balanced if the height of its left subtree is plus‐or‐minus‐one the height of its right subtree. A binary tree has the height-balanced property if all of its Nodes are balanced According to this definition, the following is NOT an AVL tree, because the root node (A) has a balance of 2. Balances of 0 are not shown. A (3‐1 = 2) B C (2‐1 = 1) D E (1‐0 = 1) Prepared By : Er. Harvinder Singh Assist Prof., CSE, H.C.T.M (Kaithal) Page ‐ 201 ‐
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1. Advanced Data Structure using C++

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