Can you define node pointers in a base binary tree class?

I've created two separate binary tree classes, with some shared functions/variables and some that are not shared. So I have tried to abstract away the similarities in a base BinaryTree class.

class BinaryTree{
public:
BinaryTree(std::vector<int> &vec);

BinaryTree *left;
BinaryTree *right;
std::vector<int> dataVector;
//functions...
}

The derived classes share the type of data with the base class (dataVector). But they require the creation of new nodes of their derived type.

class Derived : public BinaryTree{
public:
void function(){
left = new BinaryTree();
};

Obviously by creating a new BinaryTree, we lose the extended functionality of the Derived class, but if I were to declare left and right as Derived* inside the Derived class, I would defeat the whole purpose of abstraction in the first place.

How would I go about implementing this if it's even possible?

EDIT: one of the derived classes might implement quick sort

class QuickSort : public BinaryTree{ 
public:

//chooses whether to sort itself or the children nodes recursively
void sortVector(){
if(isSorting) partition(),
else if (left->isSorting || left->areNodesSorting) left->sortVector();
else if(right->isSorting || right->areNodesSorting) right->sortVector();
}

void partition(){ /* partition implementation */};

void setupNodes(){
/* choose where to split dataVector depending on the pivot, and initialize 
the pointers - left = new QuickSort(chosenVector); etc*/ };

bool isSorting = true, isLeaf = true, areNodesInit = false, areNodesSorting = 
false;

int pivotIndex, j = 0, i = -1;
int pivotValue;

}

There are obviously other helper functions. And I need step by step sorting, so I've put all the variables as instance variables - should they be part of Payload? Also as you can see some functions use recursion and need to know of all the different children nodes as well.