Help extracting nodes from binary search tree

[Skynet]

I was wondering if you could assist me with my binary search tree. I have this code so far and I want to try and make it do two more things for me. I want it to extract the smallest number in the tree with extractMin and return that value to me, then I would like for it to also have an extract function that uses extractMin if needed but I want to be able to give it a number and it looks for that number regardless where it is in the tree and simply remove it and keeps the rest of the tree intact. I have some ideas but then I realized that I would end up losing nodes or doing too much work, etc... if I were to do that. Can anyone help?

#include <iostream>
#include "binarySearchTree.h"
using namespace std;

int main()
{

	binarySearchTree tree;

	tree.insert(53);
	tree.insert(11);
	tree.insert(7);
	tree.insert(5);
	tree.insert(2);
	tree.insert(23);

	tree.display();

	tree.height();

	tree.extractmin();

	//tree.remove();
	return 0;
}

#include <iostream>
using namespace std;

class binarySearchTree
{
private:
	class node
	{
	public:
		int data;
		node * left;
		node * right;

		node(int x)
		{
			data = x;
			left = NULL;
			right = NULL;
		}
	};

	node * root;

	//recursive methods

	void recInsert(int x, node * &r)
	{
		if( r == NULL )
		{
			r = new node(x);
		}
		else
		{
			if( x < r->data )
			{
				recInsert(x, r->left);
			}
			else
			{
				recInsert(x, r->right);
			}
		}
	}

	//display items in tree rooted at r
	void recDisplay(node * r)
	{
		if( r != NULL )
		{
			recDisplay(r->left);
			cout << r->data << endl;		
			recDisplay(r->right);
		}
	}

	//return height of tree rooted at r
	int getHeight( node * r )
	{
		// Max is the greatest height below (*r), temp is just used to avoid calling getHeight(r->right) twice
		int max=0,temp=0;
		// Recurse down lhs
		if (r->left!=NULL) max=getHeight(r->left);
		// Recurse down rhs
		if (r->right!=NULL) temp=getHeight(r->right);
		if (temp>max) max=temp;

		// Return the max
		return max+1;
	}

	void extractMin(node * &r)
	{
		// Recurse down lhs
		if (r->left!=NULL) 
			extractMin(r->left);
	}

public:

	binarySearchTree()
	{
		root = NULL;
	}

	void insert(int x)
	{
		recInsert(x, root);
	}

	//display all items in tree
	void display()
	{
		recDisplay(root);
	}

	void height()
	{
		cout<<getHeight(root)<<endl;
	}

	void extractmin()
	{
		extractMin(root);
	}
};