Part A: Binary trees and Binary search trees Exercise 1: Negate a tree Given a tree of integers a, write a method that returns a new tree containing all the elements of a with their sign negated, i.e., positive integers become negative and negative integers become positive.
Exercise 2: Check for positive Given a tree of integers a, return a boolean value indicating whether all the values in its nodes are positive, i.e., >= 0.
Exercise 3: Mirror image This method should be generic in the element type E. Given a tree a, construct and return a tree that is the mirror image of a along the left-right axis.
Exercise 4: Postorder traversal Given a tree a, produce and return a list containing the values in a by trvaersing the nodes in postorder, i.e., for every node, all the values in the left subtree should be listed first, then all the values in the right subtree and then nally the value in the node itself. Hint: Recall the method for inorder traversal done in the Lecture.
Binary search trees As dissussed in the Lecture, a tree is a binary search tree if, for every node, - all the values stored in the left subtree of the node are less than the value at the node, and - all the values stored in its right subtree of the node are greater than the value at the node.
Exercise 5: Test for the search tree property Given a tree of integers a, write a method that returns a boolean value indicating whether a is a binary search tree. Hint: You may need helper functions to write this method. Document any helper functions you define.
Exercise 6: Traversing binary search trees Given a binary search tree of integers a, write a method that prints the values stored in it in descending order. Do this without building a separate list of the values.
Exercise 7: Maximum value in a search tree Assuming that the argument tree a is a binary search tree, write an efficient method to find the maximum value stored in the tree. Your method must not visit and compare all the nodes in the tree. Rather, it it must traverse at most one path in the tree from the root node. This should work in O(log n) time for a balanced binary tree. (Hint: In a binary search tree, all the values in the left subtree of a node are less than or equal to the value in the node. So, the maximum value can't be in the left subtree, right? Where can it be?)
Exercise 8: Deleting a value in a search tree Assuming that the argument tree a is a binary search tree, this method must delete the value x from a and return the resulting tree. The original tree a must not be altered. Rather, you should build a new tree that contains all the values of a except for one copy of x. The resulting tree must be again a binary search tree. Your algorithm must take time proportional to the height of the tree, which is normally O(logn).
(Hint: As discussed in Lecture, the node of x may have two subtrees. In that case, the node cannot be simply deleted. Rather,
you need to replace x in that node with the maximum value of the left subtree.)
Part B: Height-balanced trees (AVL trees) Height-balanced trees A height-balanced binary tree is a binary tree in which, for every node, the left and right subtrees have a difference in height of at most 1. Insertion/deletion in a height-balanced tree may in general destroy the height-balanced property. In that case, we can use rotations discussed in Lecture to rebalance the tree and obtain a height-balanced tree again. Please consult any Data Structures text book as well as online resoures for reference material. AVL trees A height-balanced binary search tree is called an \AVL tree" (named after its inventors, Adelson-Velsky and Landis). For checking the height-balanced property of nodes after insertions/deletions, one needs an efficient method to obtain the height of a tree. Explicit calculation requires O(n) time, which is too expensive. For this purpose, the Tree class given to you has been extended with an instance variable to store the height of the tree:
protected final int height;
It also has an instance method
public int getHeight();
that returns the stored height value.
Exercise 10: Insertion/deletion with height-balancing Write modified versions of insert and delete methods that maintain the height-balanced property of trees. You should assume that the input trees are height-balanced binary search trees and produce results that are height-balanced binary search trees. Both the methods should work in O(log n) time. You may use the isHeightBalanced method in assert statements to ensure that your code works correctly.