In deletion there is a given value x and an AVL tree T. We delete the node containing the value x and rebalance the tree if it becomes unbalance after deleting the node. So we don’t need parent pointer to travel up. Draw the result of deleting the designated value from the AVL trees shown below: a) 10 (Delete 3) 7 45 3 30 50 70 b) 10 (Delete 30) 7 45 3 30 50 70 30 (Delete 35) - 15 40 7 20 35 50 9 32 4352 30 (Delete 20) 15 40 7 20 35 50 9 32 4352 47 Minimum number of nodes in a tree with height h can be represented as: 1. AVL Trees (10 Points) Given the following AVL Tree: (a) Draw the resulting BST after 5 is removed, but before any rebalancing takes place. Draw the result of deleting the designated value from the AVL trees shown below: Good Luck 2. The AVL tree and other self-balancing search trees like Red Black are useful to get all basic operations done in O(log n) time. AVL Tree Any binary search tree that satisfies the Height-Balance property. Values are left out here, but any valid BST values could be filled in. That means that covering the basic scenarios should give you a broad coverage of the AVL tree functionality.

HW3 Solutions Page Return to homework page.

Here are some key points about AVL trees: If there are n nodes in AVL tree, minimum height of AVL tree is floor(log 2 n). So to get a minimum AVL tree of height 4, we need to build up minimum AVL trees of heights 0-3 first. Note the effects when the key node is inserted in each of the six cases. Label each node in the resulting tree with its balance factor. 4.19 Show result of inserting 2, 1, 4, 5, 9, 3, 6, 7 into an empty AVL tree.

It is complex in the sense that we may have to do more than one rotations to rebalance the tree after deleting a node.

In exercise 13.3-2 (problem 12), you found the red-black tree that results from successively inserting the keys 41, 38, 31, 12, 19, and 8 into an initially empty tree. Solution : Deleting 55 from the AVL Tree disturbs the balance factor of the node 50 i.e. The recursive code itself travels up and visits all the ancestors of the deleted node.

minimum AVL tree of height h-1, and the other a minimum AVL tree of h-2. Re-balance = re-organize the nodes in an out-of-balanced AVL tree so that the resulting tree will satisfy the height constraint of an AVL tree (I.e., the result of the re-balance operation is an AVL tree 2. Now show the red-black trees that result from the successive deletion of the keys in the order 8, 12, 19, 31, 38, and 41. Delete Node 55 from the AVL tree shown in the following image. Practical session Practical session No. AVL trees are often compared with red–black trees because both support the same set of operations and take ... To split an AVL tree into two smaller trees, those smaller than key x, and those larger than key x, first draw a path from the root by inserting x into the AVL.

The action position is a reference to the parent node from which a node has been physically removed. 2 of 12 (b) Now rebalance the tree that results from (a). Draw the result of deleting the designated value from the AVL trees shown below: The AVL trees are more balanced compared to Red-Black Trees, but they may cause more rotations during insertion and deletion. The action position indicate the first node whose height has been affected (possibly changed) by the deletion (This will be important in the re-balancing phase to adjust the tree back to an AVL tree) An Example Tree that is an AVL Tree The above tree is AVL because differences between heights of left and right subtrees for every node is less than or equal to 1.

Thus, it has Θ(logn) height, which implies Θ(logn) worst case search and insertion times. If there are n nodes in AVL tree, maximum height can’t exceed 1.44*log 2 n. If height of AVL tree is h, maximum number of nodes can be 2 h+1 – 1.

In the recursive BST delete, after deletion, we get pointers to all ancestors one by one in bottom up manner.

The image below shows each of these, and finally a minimum AVL tree of height 4. No. Following is the C implementation for AVL Tree Deletion. 100 55 155 25 100 5 155 25 100 55 155 255 200 100 5 200

Inserting the first value. Thus, once again, as above, to restructure the tree after a delete we will call the restructure method on the parent of the deleted node.

2.

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