Step $4$: Implement ExtendedAVLTree and ExtendedAVLNode

Each node in an ExtendedAVLTree must have a correct subtreeKeyCount after an insertion or removal operation. Determine which member functions in AVLTree and AVLNode must be overridden in ExtendedAVLTree and ExtendedAVLNode to keep each node's subtreeKeyCount correct. New functions can be added along with overridden functions, if desired.

Hint: Consider an UpdateSubtreeKeyCount$()$ member function for the ExtendedAVLNode class. The function requires each child node's subtreeKeyCount to be correct, and updates the node's subtreeKeyCount appropriately. Overridden functions in both ExtendedAVLNode and ExtendedAVLTree can call a node's UpdateSubtreeKeyCount$()$ function as needed.

Once determinations are made, complete the implementation of both the ExtendedAVLTree and ExtendedAVLNode classes. Do not implement ExtendedAVLTree's GetNthKey$()$ in this step. GetNthKey$()$ requires correct subtree counts at each node.

Step $5$: Run tests

TreeTestCommand is an abstract base class defined in TreeCommands.h$.$ A TreeTestCommand object is an executable command that operates on a binary search tree. Classes inheriting from TreeTestCommand are also declared in TreeCommands.h:

TreeInsertCommand inserts keys into the tree

TreeRemoveCommand removes keys from the tree

TreeVerifyKeysCommand verifies the tree's keys using an inorder traversal

TreeVerifySubtreeCountsCommand verifies that each node in the tree has the expected subtree key count

TreeGetNthCommand verifies that GetNthKey$()$ returns an expected value

Code in main.cpp contains three automated test cases. Each test executes a vector of TreeTestCommand objects in sequence. The third test includes TreeGetNthCommands and will not pass until the completion of Step $6.$ The first two tests should pass after completion of step $4.$

Before proceeding to Step $6,$ ensure that the first two tests in main.cpp pass.

Step $6$: Implement ExtendedAVLTree's GetNthKey$()$ member function $($worst case O$($log n$))$

GetNthKey$()$ must return the tree's nth$-$largest key. The parameter n starts at $0$ for the smallest key in the tree. Ex: Suppose a tree has keys:

$10,19,20,30,42,55,77$

Then GetNthKey$(0)$ returns $10,$ GetNthKey$(1)$ returns $19,...,$ GetNthKey$(5)$ returns $55,$ and GetNthKey$(6)$ returns $77.$

Determine an algorithm that uses the subtree key counts so that GetNthKey$()$ operates in worst case O$($log n$)$ time.

Here are the files you'll need. You should not need to examine the contents of the last three. They are for testing purposes only.