In CSC 321, we considered basic implementations of binary trees and binary search trees. The following classes are variations of the versions from last semester: TreeNode.java, BinaryTree.java, BinarySearchTree.java. The BinaryTree class contains several methods that use a divide-and-conquer approach, including an O(N) size method. The BinarySearchTree class, which inherits from BinaryTree, overrides the add and contains methods with O(log N) decrease-and-conquer versions. It also contains inefficient O(N) implementations of select and rank, which you will similarly replace with O(log N) decrease-and-conquer versions.

Part 1: O(N) asList

Currently, the asList method is incomplete. Implement this method using a divide-and-conquer approach. In particular, start with an empty list and then recursively traverse the tree (using an inorder traversal), adding each data item to the list. Finally, return that list. Note that this method is used by the toString method, so it should be possible to print the contents of a binary tree after you have completed the method.

Part 2: O(1) size

Currently, the size method uses divide-and-conquer recursion to traverse the tree and count the number of nodes. To avoid the O(N) cost of this approach, you are to modify TreeNode so that in addition to storing the data value and subtrees, each node also stores the sizes of its subtrees (and those values are updated when either subtree is changed). With this size information embedded in each node, it becomes simple to determine the size of any subtree in a tree. For example, suppose current referred to a node in a binary tree. if you knew that the left subtree of current contained 12 nodes, and the right subtree of current contained 14 nodes, then the tree rooted at current would contain 12+14+1 = 27 nodes.

Note: all of the TreeNode methods should remain O(1).

Once you have made your modifications, reimplement the size method in BinaryTree so that it utilizes the size information in nodes to determine the tree size in O(1) time.

Part 3: O(log N) select

Currently, the select method in BinarySearchTree is O(N), since it first converts the tree into a list (using your asList method) and then accesses the corresponding index in that list. Alternatively, a decrease-and-conquer approach can be taken, similar to the approach used in the quick-select algorithm described in lectures. Consider the following binary search tree:

Suppose you want the rank-4 value in the tree. Since there are 3 values in the left subtree, the rank-4 value must be at the root. If you wanted the rank-3 value in the tree, it must be in the left subtree (and in fact is the rank-3 value in the left subtree). If you wanted the rank-5 value in the tree, it must be in the right subtree (and in fact is the rank-1 value in the right subtree).

You are to reimplement the select operation using this decrease-and-conquer approach, which does work proportional to the height of the binary search tree. As long as the tree is relatively balanced, the select operation will be O(log N).

Part 4: O(log N) rank

Similarly, it is possible to utilize the size information in nodes to implement an O(log N) rank operation. The rank operation is the inverse of select - it finds a specific value in the tree and returns its order rank. For example, the rank of "cubs" in the above binary search tree is 2, whereas the rank of "pirates" is 5. If the rank method is called on a value that does not appear in the tree, it should return -1.

You are to reimplement the rank method using a decrease-and-conquer approach, so that the amount of work is proportional to the height of the binary search tree.