Design an algorithm for the following operations for a binary tree BT, and show the worst-case running times for each implementation

computer science

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1. (15 marks) Design an algorithm for the following operations for a binary tree BT, and show the worst-case running times for each implementation: preorderNext(x): return the node visited after node x in a pre-order traversal of BT. postorderNext(x): return the node visited after node x in a post-order traversal of BT. inorderNext(x): return the node visited after node x in an in-order traversal of BT. 

2. (25 marks) Design a recursive linear-time algorithm that tests whether a binary tree satisfies the search tree order property at every node. 3. (20 marks) Exercise 8.


2. Illustrate what happens when the sequence 1, 5, 2, 4, 3 is added to an empty ScapegoatTree, and show where the credits described in the proof of Lemma 8.

3 go, and how they are used during this sequence of additions. 

4. (20 marks) Implement a commonly used hash table in a program that handles collision using linear probing. Using (K mod 13) as the hash function, store the following elements in the table: {1, 5, 21, 26, 39, 14, 15, 16, 17, 18, 19, 20, 111, 145, 146}. 5. (20 marks) Exercise 6.

7. Create a subclass of BinaryTree whose nodes have fields for storing preorder, post-order, and in-order numbers. Write methods preOrderNumber(), inOrderNumber(), and postOrderNumbers() that assign these numbers correctly. These methods should each run in O(n) time.

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