Using the following c++ header file near the bottom for context, fill in the "TODO" commented parts of the following function to finish the function implementation for the following DeleteVertex function: FUNCTION: // PURPOSE: Deletes a vertex that has E as its element. // PARAM: element E to be removed // ALGORITHM: First we must find the vertex then call Remove void BST::DeleteVertex(el_t E) { cout << "Trying to delete " << E << endl; Vertex *V; // the current vertex Vertex *Parent; // Parent will always point to V's parent // case 1: Lonely Root -------------------- if ((E == root->elem) && (root->left == NULL) && (root->right == NULL)) { cout << "...deleting the lonely root" << endl; delete root; root = NULL; return; } // only the Root was there and deleted it // case 2: One Substree Root ---------------- // if what you want to delete is the root if (/*TODO*/) { cout << "... deleting the root with just one child" << endl; if (/*TODO*/) { // TODO if it has only the left subtree // change the root to the left child and return // making sure the old root has been deleted and the new root's UP is NULL } else if (/*TODO*/) { // if it has only the right subtree // TODO change the root to the right child and return // making sure the old root has been deleted and the new root's UP is NULL } }// end of deleting the root with one subtree // ---- Otherwise deleting something else -------------- V = root; // start with the root to look for E Parent = NULL; // above the V so does not point to anything yet // going down the tree looking for E while (V != NULL) { if (E == V->elem) { // found it cout << "...removing " << V->elem << endl; // TODO call remove here to remove V return; } else if (E < V->elem) { cout << "...going to the left" << endl; // TODO update Parent and V here to go down } else { cout << "...going to the right" << endl; // TODO update Parent and V here to go down } }// end of while // reached NULL -- did not find it cout << "Did not find the key in the tree." << endl; }// end of DeleteVertex ----------------------------------------------------------------------------------------------------------- HEADER FILE:// tree element type is int for nowtypedef int el_t; // el_t is hidden from the client // definition of what a Vertex is - also hidden from the clientstruct Vertex{ Vertex *up; // points to the parent node Vertex *left; el_t elem; Vertex *right; int height; int balance;}; // this is set up to be inherited by another classclass BST {public: BST(); // intializes Root ~BST(); // destructor calls dtraverse to destroy the dynamic tree // PURPOSE: these will show the vertices in IN order // TO CALL: No parameter but provide a pointer to // the root vertex in calling INorderTraversal void Display(); void InOrderTraversal(Vertex*); // recursive // PURPOSE: these will search in PRE order - same as Depth First // TO CALL: provide the element to search for; provide a pointer to // the root vertex in calling PREorderSearch bool Search(el_t); bool PreOrderSearch(Vertex*, el_t); // recursive

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Using the following c++ header file near the bottom for context, fill in the "TODO" commented parts of the following function to finish the function implementation for the following DeleteVertex function:

FUNCTION:

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HEADER FILE:
// tree element type is int for now
typedef int el_t; // el_t is hidden from the client

// definition of what a Vertex is - also hidden from the client
struct Vertex
{
Vertex *up; // points to the parent node
Vertex *left;
el_t elem;
Vertex *right;
int height;
int balance;
};

// this is set up to be inherited by another class
class BST {
public:
BST(); // intializes Root
~BST(); // destructor calls dtraverse to destroy the dynamic tree

// PURPOSE: these will show the vertices in IN order
// TO CALL: No parameter but provide a pointer to
// the root vertex in calling INorderTraversal
void Display();
void InOrderTraversal(Vertex*); // recursive

// PURPOSE: these will search in PRE order - same as Depth First
// TO CALL: provide the element to search for; provide a pointer to
// the root vertex in calling PREorderSearch
bool Search(el_t);
bool PreOrderSearch(Vertex*, el_t); // recursive