2. The following keys are inserted into an initially empty B-Tree of order 3. Construct the B-Tree 56, 12, 32, 6, 37, 85, 79, 99. b. Then delete the nodes in the following order: 99, 37 and 6. :

11. Preorder Traversal What is the preorder traversal order of the following tree: 3 Pick ONE option 1, 2, 4, 5, 3 4, 2, 5, 1, 3 4,5, 2, 3, 1 1,2, 3, 4, 5 Clear Selection

a) A list of student's names is maintained in an AVL tree T. Design an algorithm for performing the operation findAll to return all the names in the AVL tree that match the searched name. Note that there is a possibility that same names exist in the tree. In the case that the same name exists, the name is added to the left subtree. b) Delete element 60 in the following Binary Search Tree (BST). Provide a brief description/explanation how the delete operation is performed. Your explanation can be in point form. Show the final Binary Search Tree (after the deletion of element 60). 10 20 ส 25 32 30 40 49 45 50 180 55 51 60 230 70 74 80 75 76

Exercise 3: Multi-way Trees A way to reduce the height of tree and ensure balance is to allow multiple children of nodes. In your class you learned 2-3 trees which allows up to 2 keys in a node, and the number of children is equal to the number of keys + 1. B-trees extend this concept to any arbitrary number of keys (usually number of keys is even and number of children (equal to number of keys+1) is odd). Assume we want to design a 5-way B-Tree. This will mean that there can be maximum 4 keys in a node, and if the number of keys becomes 5, we can split it into two (the same way we split 2-3 tree when number of keys becomes 3). Design a 5-way B-tree. Starting with an empty tree, insert the following keys in order. 2, 3, 5, 7, 10, 50, 22, 44, 45, 55, 66, 68, 70, 17, 6, 21, 67

The next few questions refer to the following tree: (Tree 2) 46 23 (21 29 (48 (89 (26 40 Question 5 Referring to tree 2 shown above: In which order would the nodes of the tree be visited by an in-order traversal? O 5, 21, 26, 40, 29, 23, 48, 89, 71, 46 O 5, 21, 23, 26, 29, 40, 46, 48, 71, 89 O 46, 23, 71, 21, 29, 48, 89, 5, 26, 40 O 46, 23, 21, 5, 29, 26, 40, 71, 48, 89

14- Suppose that T is a binary tree with 14 nodes. What is the minimum possible depth of T? а.) 0 b.) 3 с.) 4 d.) 5

2. This question is about binary search tree.a. Show the final result after inserting 3, 1, 4, 6, 9, 2, 8, 5, 7, 0 into an empty binary search tree.b. Show the result of deleting the root twice. If the deleted node contains 2 children, its value is replaced by the minimum value of the right subtree, and the deletion is propagated. Show the result after each deletion

Construct AVL trees by inserting the elements successively, starting with the empty tree. a) 1, 2, 3, 4, 5, 6 b) 6, 5, 4, 3, 2, 1 c) 3, 6, 5, 1, 2, 4 (Note: Observe the final trees and you will find that although all of them have the same data, the tree are not necessarily the same.)

Operations on B-Trees: a. Construct a B-Tree by using 4, 2, 3, 9, 7, 8, 6, 11, 12, 1 having the order of 3. Show all the required steps clearly.  b. Delete the element 8 from your B-Tree. Show all the steps clearly till you get the resultant tree after the deletion operation.  c. What are the time complexities of B-Tree insertion and deletion operations?  Explain how you achieved the answer.

a) Show the BST that is created by inserting the following numbers in the given order 80, 100, 40, 105, 90, 30, 20, 35, 38, 32,25, 10 b) Delete 40 from the tree c) Give the PreOrder Traversal of the tree in (a) d) Give the PostOrder Traversal of the tree in (a) e) What is the runtime of PostOrder Traversal of the tree in (a)?