Population

AVLTree(with remove)

    A. Second Edition

    This is second edition of my AVL Tree and the reason I restart this project is that I was blamed for not finishing

    remove function. So, let's finish it and it is so "Ad-Hoc".

    B.The problem

    To write AVL tree on template basis and try to keep as much as possible of original BST frame work

    because the code by Mr. Shaffer is very concise and compact! And efficiency is also a very important

    issue here. As AVLTree has to store extra information than a BST, it is expected that we need to

    reduce as many "balancing operations" as possible.

C.The idea of program
¡¡

The main idea is similar to "insert" except that the trouble node is not always at same side as

the removed node, while for inserting, it is always the case. One more problem is that the son node

of the "Axis" node can have a balance factor of 0! What's worse, his brother may also has 0 as its

balance factor while their parent has a +/-2 as factor! What should we do about it? It seems there

is no algorithm about remove. Should I check "data structure" text book for confirmation?

    D.The major functions

    1. bool insert(const Elem& e)

    Do you expect that I might start from here? But no, I didn't change anything here. And it is only

    after I finished, I thought I can omit it even "inserthelp" is not virtual.

    2. BinNode<Elem>* inserthelp(BinNode<Elem>*, const Elem&);

    This function is almost same as original BST except I try to update the height after each insertion

    which will go up from the inserted new leaf along path. And before that insertion, I placed a road

    sign "inLeaf" to indicate which side the path takes.

    3. void updateHeight(BinNode<Elem>*& subroot);

    This is the key part of program where you update height first and then try to examine the balance

    and try to keep it. It is the tricky part as I change the code many times. Finally I realized that

    there are two big cases: a) The first root which is also the first node with factor 2/-2; b) The node

    whose son node has factor of 2/-2; There are some extra conditions to examine the "first" in a) to

    make sure it is the "first".

    4. int getTreeHeight(BinNode<Elem>* subroot);

    I resist to use recursive method because the field "height" is a short-cut.

    5. BinNode<Elem>* singleRotate(BinNode<Elem>* parent, bool isRoot, bool left2right);

    Don't forget to adjust balance after rotating and the sequence is important as you have to do it from

    bottom-up.

    6. BinNode<Elem>* doubleRotate(BinNode<Elem>* parent, bool isRoot, bool left2right);

    I made it look simple by adding the "doDouble" and quite satisfy with it.

    E.Further improvement

    ¡¡

    F.File listing

    1. AVLTree.h

    2. BinNode.h

    3. BST.h

    4. dictionary.h

    5. Elem.h

    6. AVLTree.cpp  (main)

    ¡¡

    file name: BinNode.h

    // Binary tree node abstract class
    template <class Elem> class BinNode {
    public:
    	// Return the node's element
    	virtual Elem& val() = 0;
    	// Set the node's element
    	virtual void setVal(const Elem&) = 0;
    	// Return the node's left child
    	virtual BinNode* left() const = 0;
    	// Set the node's left child
    	virtual void setLeft(BinNode*) = 0;
    	// Return the node's right child
    	virtual BinNode* right() const = 0;
    	// Set the node's right child
    	virtual void setRight(BinNode*) = 0;
    	// Return true iff the node is a leaf
    	virtual bool isLeaf() = 0;
    	//my personal preference
    	virtual BinNode<Elem>* getSon(bool isLeft)const=0;
    	//my personal preference
    	virtual void setSon(BinNode<Elem>* son, bool isLeft)=0;
    };

    // Binary tree node class
    template <class Elem>
    class BinNodePtr : public BinNode<Elem> {
    private:
    	Elem it;                     // The node's value
    	BinNodePtr* lc;              // Pointer to left child
    	BinNodePtr* rc;              // Pointer to right child
    public:
    	// Two constructors -- with and without initial values
    	BinNodePtr() { lc = rc = NULL; }
    	BinNodePtr(Elem e, BinNodePtr* l =NULL,
    					 BinNodePtr* r =NULL)
    	{ it = e; lc = l; rc = r; }
    	~BinNodePtr() {}             // Destructor
    	Elem& val() { return it; }
    	void setVal(const Elem& e) { it = e; }
    	inline BinNode<Elem>* left() const { return lc; }
    	void setLeft(BinNode<Elem>* b) { lc = (BinNodePtr*)b; }
    	inline BinNode<Elem>* right() const { return rc; }
    	void setRight(BinNode<Elem>* b) { rc = (BinNodePtr*)b; }
    	bool isLeaf() { return (lc == NULL) && (rc == NULL); }
    	BinNode<Elem>* getSon(bool isLeft)const {return isLeft?lc:rc;}
    	void setSon(BinNode<Elem>* son, bool isLeft)
    	{
    		isLeft?setLeft(son):setRight(son);
    	}
    };

    template <class Elem>
    class AVLNodePtr : public BinNode<Elem>
    {
    protected:
    	Elem it;                     // The node's value
    	AVLNodePtr* lc;              // Pointer to left child
    	AVLNodePtr* rc;              // Pointer to right child
    	int height;
    	bool inLeft;
    public:
     // Two constructors -- with and without initial values
    	AVLNodePtr() { lc = rc = NULL; height=1; inLeft=true; }
    	AVLNodePtr(Elem e, AVLNodePtr<Elem>* l =NULL,
    					 AVLNodePtr<Elem>* r =NULL, int newHeight=1)
    	{ it = e; lc = l; rc = r; height=newHeight; inLeft=true;}
    	~AVLNodePtr() {}             // Destructor
    	Elem& val() { return it; }
    	void setVal(const Elem& e) { it = e; }
    	BinNode<Elem>* left() const { return lc; }
    	void setLeft(BinNode<Elem>* b) { lc = (AVLNodePtr*)b; }
    	inline BinNode<Elem>* right() const { return rc; }
    	void setRight(BinNode<Elem>* b) { rc = (AVLNodePtr*)b; }
    	bool isLeaf() { return (lc == NULL) && (rc == NULL); }
    	void setHeight(int newHeight){height=newHeight;}
    	int getHeight(){return height;}
    	BinNode<Elem>* getSon(bool isLeft)const {return isLeft?lc:rc;}
    	bool getSide() const {return inLeft;}
    	void setSide(bool isLeft){ inLeft=isLeft;}
    	void setSon(BinNode<Elem>* son, bool isLeft)
    	{
    		isLeft?setLeft(son):setRight(son);
    	}

    };

    ¡¡

    file name: BST.h

    // This file includes all of the pieces of the BST implementation

    #include "dictionary.h"

    #include "binnode.h"

    // Binary Search Tree implementation for the Dictionary ADT
    template <class Key, class Elem, class KEComp, class EEComp>
    class BST : public Dictionary<Key, Elem, KEComp, EEComp> {
    protected:
      BinNode<Elem>* root;   // Root of the BST
      int nodecount;         // Number of nodes in the BST
      // Private "helper" functions
      void clearhelp(BinNode<Elem>*);
      BinNode<Elem>* inserthelp(BinNode<Elem>*, const Elem&);
      BinNode<Elem>* deletemin(BinNode<Elem>*, BinNode<Elem>*&);
      BinNode<Elem>* removehelp(BinNode<Elem>*, const Key&,
                                BinNode<Elem>*&);
      bool findhelp(BinNode<Elem>*, const Key&, Elem&) const;
      void printhelp(BinNode<Elem>*, int) const;
    public:
      BST() { root = NULL; nodecount = 0; }  // Constructor
      ~BST() { clearhelp(root); }            // Destructor
      void clear()
        { clearhelp(root); root = NULL; nodecount = 0; }
      bool insert(const Elem& e) {
        root = inserthelp(root, e);
        nodecount++;
        return true;
      }
      bool remove(const Key& K, Elem& e) {
        BinNode<Elem>* t = NULL;
        root = removehelp(root, K, t);
        if (t == NULL) return false;  // Nothing found
        e = t->val();
        nodecount--;
        delete t;
        return true;
      }
      bool removeAny(Elem& e) { // Delete min value
        if (root == NULL) return false; // Empty tree
        BinNode<Elem>* t;
        root = deletemin(root, t);
        e = t->val();
        delete t;
        nodecount--;
        return true;
      }
      bool find(const Key& K, Elem& e) const
        { return findhelp(root, K, e); }
      int size() { return nodecount; }
      void print() const {
        if (root == NULL) cout << "The BST is empty.\n";
        else printhelp(root, 0);
      }
    };

    template <class Key, class Elem, class KEComp, class EEComp>
    void BST<Key, Elem, KEComp, EEComp>::
    clearhelp(BinNode<Elem>* subroot) {
      if (subroot == NULL) return;
      clearhelp(subroot->left());
      clearhelp(subroot->right());
      delete subroot;
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    BinNode<Elem>* BST<Key, Elem, KEComp, EEComp>::
    inserthelp(BinNode<Elem>* subroot, const Elem& val) {
      if (subroot == NULL)            // Empty tree: create node
        return (new BinNodePtr<Elem>(val, NULL, NULL));
      if (EEComp::lt(val, subroot->val())) // Insert on left
        subroot->setLeft(inserthelp(subroot->left(), val));
      else subroot->setRight(inserthelp(subroot->right(), val));
      return subroot;  // Return subtree with node inserted
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    BinNode<Elem>* BST<Key, Elem, KEComp, EEComp>::
    deletemin(BinNode<Elem>* subroot, BinNode<Elem>*& min) {
      if (subroot->left() == NULL) { // Found min
        min = subroot;
        return subroot->right();
      }
      else {                         // Continue left
        subroot->setLeft(deletemin(subroot->left(), min));
        return subroot;
      }
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    BinNode<Elem>* BST<Key, Elem, KEComp, EEComp>::
    removehelp(BinNode<Elem>* subroot, const Key& K,
               BinNode<Elem>*& t) {
      if (subroot == NULL) return NULL; // Val is not in tree
      else if (KEComp::lt(K, subroot->val())) // Check left
        subroot->setLeft(removehelp(subroot->left(), K, t));
      else if (KEComp::gt(K, subroot->val())) // Check right
        subroot->setRight(removehelp(subroot->right(), K, t));
      else {                           // Found it: remove it
        BinNode<Elem>* temp;
        t = subroot;
        if (subroot->left() == NULL)       // Only a right child
          subroot = subroot->right();      //  so point to right
        else if (subroot->right() == NULL) // Only a left child
          subroot = subroot->left();       //  so point to left
        else {                    // Both children are non-empty
          subroot->setRight(deletemin(subroot->right(), temp));
          Elem te = subroot->val();
          subroot->setVal(temp->val());
          temp->setVal(te);
          t = temp;
        }
      }
      return subroot;
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    bool BST<Key, Elem, KEComp, EEComp>:: findhelp(
          BinNode<Elem>* subroot, const Key& K, Elem& e) const {
      if (subroot == NULL) return false;         // Empty tree
      else if (KEComp::lt(K, subroot->val()))    // Check left
        return findhelp(subroot->left(), K, e);
      else if (KEComp::gt(K, subroot->val()))    // Check right
        return findhelp(subroot->right(), K, e);
      else { e = subroot->val();  return true; } // Found it
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    void BST<Key, Elem, KEComp, EEComp>::
    printhelp(BinNode<Elem>* subroot, int level) const {
      if (subroot == NULL) return;          // Empty tree
      printhelp(subroot->left(), level+1);  // Do left subtree
      for (int i=0; i<level; i++)           // Indent to level
        cout << "  ";
      cout << subroot->val() << "\n";       // Print node value
      printhelp(subroot->right(), level+1); // Do right subtree
    }

    ¡¡

    file name: dictionary.h

    ¡¡

    // The Dictionary abstract class.  KEComp compares a key
    // and an element. EEComp compares two elements.
    template <class Key, class Elem, class KEComp, class EEComp>
    class  Dictionary {
    public:
      // Reinitialize dictionary
      virtual void clear() = 0;
      // Insert an element.  Return true if insert is
      // successful, false otherwise.
      virtual bool insert(const Elem&) = 0;
      // Remove some element matching Key.  Return true if such
      // exists, false otherwise.  If multiple entries match
      // Key, an arbitrary one is removed.
      virtual bool remove(const Key&, Elem&) = 0;
      // Remove and return an arbitrary element from dictionary.
      // Return true if some element is found, false otherwise.
      virtual bool removeAny(Elem&) = 0;
      // Return a copy of some Elem matching Key.  Return true
      // if such exists, false otherwise.  If multiple elements
      // match Key, return an arbitrary one.
      virtual bool find(const Key&, Elem&) const = 0;
      // Return the number of elements in the dictionary.
      virtual int size() = 0;
    };

    ¡¡

    file name: Elem.h

    //This is the element of login system

    class KEComp
    {
    public:
    	static bool lt(int key, int elem) {return key<elem;}
    	static bool eq(int key, int elem) {return key==elem;}
    	static bool gt(int key, int elem) {return key>elem;}
    };

    class EEComp
    {
    public:
    	static bool lt(int e1, int e2)
    		{ return e1<e2;}
    	static bool eq(int e1, int e2)
    		{ return e1==e2;}
    	static bool gt(int e1, int e2)
    		{ return e1>e2;}
    };

    ¡¡

    file name:  AVLTree.h

    #include "BST.h"

    template<class Elem>
    struct Descriptor
    {
    	BinNode<Elem>* parent;
    	bool isRoot;
    	bool isLeft;
    	bool isSingle;
    	bool left2right;
    };

    template<class Key, class Elem, class KEComp, class EEComp>
    class AVL: public BST<Key, Elem, KEComp, EEComp>
    {
    protected:
    //	BinNode<Elem>* startPtr;
    	void clearhelp(BinNode<Elem>*);
    	virtual BinNode<Elem>* inserthelp(BinNode<Elem>*, const Elem&);
    	bool findhelp(BinNode<Elem>*, const Key&, Elem&) const;
    	void printhelp(BinNode<Elem>*, int) const;
    	void updateHeight(BinNode<Elem>*& subroot);
    	int  getFactor(BinNode<Elem>* subroot);
    	void adjust(BinNode<Elem>*& subroot, bool isRoot);
    	int getTreeHeight(BinNode<Elem>* subroot);
    	void adjustHeight(BinNode<Elem>* subroot);
    	BinNode<Elem>* singleRotate(BinNode<Elem>* parent, bool isRoot, bool left2right);
    	BinNode<Elem>* doubleRotate(BinNode<Elem>* parent, bool isRoot, bool left2right);
    	void checkDir(BinNode<Elem>* subroot, bool& isSingle, bool& left2right);
    	BinNode<Elem>* doDouble(BinNode<Elem>* oldRoot, bool left2right);
    	virtual BinNode<Elem>* deletemin(BinNode<Elem>*, BinNode<Elem>*&);
    	virtual BinNode<Elem>* removehelp(BinNode<Elem>*, const Key&, BinNode<Elem>*&);
    public:
    	AVL() { root = NULL; nodecount = 0; }  // Constructor
    	~AVL() { clearhelp(root); root=NULL; }            // Destructor
    	/*
    	bool insert(const Elem& e)
    	{
    		root = inserthelp(root, e);
    		nodecount++;
    		return true;
    	}
    	*/
    };

    //do not use recursive!!!!!
    template <class Key, class Elem, class KEComp, class EEComp>
    int AVL<Key, Elem, KEComp, EEComp>::getTreeHeight(BinNode<Elem>* subroot)
    {
    	AVLNodePtr<Elem>* ptr, *l, *r;
    	int newHeight, lHeight, rHeight;//, factor;//, sonFactor;

    	if (subroot==NULL)
    	{
    		return 0;
    	}

    	ptr=(AVLNodePtr<Elem>*)subroot;
    	l=(AVLNodePtr<Elem>*)ptr->left();
    	r=(AVLNodePtr<Elem>*)ptr->right();
    	if (l==NULL)
    	{
    		lHeight=0;
    	}
    	else
    	{
    		lHeight=l->getHeight();
    	}
    	if (r==NULL)
    	{
    		rHeight=0;
    	}
    	else
    	{
    		rHeight=r->getHeight();
    	}
    	newHeight=1+(lHeight>rHeight?lHeight:rHeight);
    	return newHeight;
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    void AVL<Key, Elem, KEComp, EEComp>::adjustHeight(BinNode<Elem>* subroot)
    {
    	int height;
    	if (subroot==NULL)
    	{
    		return;
    	}
    	height=getTreeHeight(subroot);
    	((AVLNodePtr<Elem>*)(subroot))->setHeight(height);
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    BinNode<Elem>* AVL<Key, Elem, KEComp, EEComp>::doDouble(BinNode<Elem>* oldRoot,
    														bool left2right)
    {
    	BinNode<Elem> *small, *mid, *big,*t1,*t2,*t3,*t4;
    	if (left2right)
    	{
    		big=oldRoot;//the root;
    		small=oldRoot->left();
    		mid=small->right();
    		t1=small->left();
    		t2=mid->left();
    		t3=mid->right();
    		t4=big->right();
    	}
    	else
    	{
    		small=oldRoot;
    		big=small->right();
    		mid=big->left();
    		t1=small->left();
    		t2=mid->left();
    		t3=mid->right();
    		t4=big->right();
    	}
    	mid->setLeft(small);
    	mid->setRight(big);
    	small->setLeft(t1);
    	small->setRight(t2);
    	big->setLeft(t3);
    	big->setRight(t4);
    	adjustHeight(small);
    	adjustHeight(big);
    	adjustHeight(mid);
    	return mid;
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    BinNode<Elem>* AVL<Key, Elem, KEComp, EEComp>::doubleRotate(BinNode<Elem>* parent,
    	bool isRoot, bool left2right)
    {
    	BinNode<Elem>* oldRoot;
    	bool isLeft;

    	if (isRoot)
    	{
    		oldRoot=parent;
    		root=doDouble(oldRoot, left2right);

    		return root;//because we need parent return as real root
    	}
    	else
    	{
    		isLeft=((AVLNodePtr<Elem>*)parent)->getSide();
    		oldRoot=parent->getSon(isLeft);
    		parent->setSon(doDouble(oldRoot, left2right), isLeft);
    		adjustHeight(parent);
    		return parent;
    	}
    }

    //if isRoot, we don't need isLeft, it is useful when it is not root and
    //we need to know which son it is in
    template <class Key, class Elem, class KEComp, class EEComp>
    BinNode<Elem>* AVL<Key, Elem, KEComp, EEComp>::singleRotate(BinNode<Elem>* parent,
    	bool isRoot, bool left2right)
    {
    	BinNode<Elem>* oldRoot=parent, *son, *t1;
    	bool isLeft=((AVLNodePtr<Elem>*)parent)->getSide();

    	if (isRoot)
    	{
    		son=parent->getSon(left2right);
    		t1=son->getSon(!left2right);
    		son->setSon(parent, !left2right);
    		parent->setSon(t1, left2right);
    		adjustHeight(parent);//sequence is VERY IMPORTANT!
    		adjustHeight(son);//sequence is VERY IMPORTANT!

    		root=son;
    		return son;//because now, we need return son as parent;
    		/*
    		son=parent->getSon(isLeft);
    		t1=son->getSon(!left2right);
    		son->setSon(parent, !left2right);
    		parent->setSon(t1, left2right);
    		//because parent is at lower level now, we need adjust parent first!!!
    		adjustHeight(parent);//sequence is VERY IMPORTANT!
    		adjustHeight(son);//sequence is VERY IMPORTANT!

    		root=son;
    		return son;//because now, we need return son as parent;
    		*/
    	}
    	else
    	{
    		//for non-root rotation, parent doesn't change!!!!!
    		//it is now very concise!!
    		oldRoot=parent->getSon(isLeft);
    		son=oldRoot->getSon(left2right);//this is the trick!
    		t1=son->getSon(!left2right);
    		parent->setSon(son, isLeft);
    		oldRoot->setSon(t1, left2right);
    		son->setSon(oldRoot, !left2right);
    		//sequence is very important!!!
    		adjustHeight(oldRoot);
    		adjustHeight(son);
    		adjustHeight(parent);//adjust sequence: from low to high!!!
    		return parent;
    	}

    }

    //check the direction of rotation by returning value in reference
    template<class Key, class Elem, class KEComp, class EEComp>
    void AVL<Key, Elem, KEComp, EEComp>::checkDir(BinNode<Elem>* subroot,
    											  bool& isSingle, bool& left2right)
    {
    	BinNode<Elem>* son;
    	int parentFactor, sonFactor;
    	bool isLeft;
    	isLeft=((AVLNodePtr<Elem>*)subroot)->getSide();
    	son=subroot->getSon(isLeft);
    	parentFactor=getFactor(subroot);
    	//to do
    	sonFactor=getFactor(son);
    	if (sonFactor==0)
    	{
    		son=subroot->getSon(!isLeft);
    		sonFactor=getFactor(son);
    		if (sonFactor==0)
    		{
    			isSingle=true;
    			left2right=parentFactor>0;
    			return;
    		}
    	}
    	isSingle=parentFactor*sonFactor>0;//same sign
    	left2right=parentFactor>0;
    }

    //if isroot, isLeft will be ignored.
    template <class Key, class Elem, class KEComp, class EEComp>
    void AVL<Key, Elem, KEComp, EEComp>::adjust(BinNode<Elem>*& subroot, bool isRoot)
    {
    	BinNode<Elem>* temp;
    	bool isSingle, left2right, isLeft;
    	if (isRoot)
    	{
    		temp=subroot;
    	}
    	else
    	{
    		//use its son to check
    		isLeft=((AVLNodePtr<Elem>*)subroot)->getSide();
    		temp=subroot->getSon(isLeft);
    		/*
    		if (getFactor(temp)==0)
    		{
    			temp=subroot->getSon(!isLeft);
    		}
    		*/
    	}

    	checkDir(temp, isSingle, left2right);
    	if (isSingle)
    	{
    		//it helps reading and for singleRotate isLeft is ignored when it is isRoot
    		subroot=singleRotate(subroot, isRoot, left2right);
    	}
    	else
    	{
    		subroot=doubleRotate(subroot, isRoot, left2right);
    	}
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    int AVL<Key, Elem, KEComp, EEComp>::getFactor(BinNode<Elem>* subroot)
    {
    	int lHeight, rHeight;
    	AVLNodePtr<Elem>* ptr, *l, *r;
    	if (subroot==NULL)
    	{
    		return 0;
    	}
    	ptr=(AVLNodePtr<Elem>*)subroot;
    	l=(AVLNodePtr<Elem>*)(ptr->left());
    	r=(AVLNodePtr<Elem>*)(ptr->right());
    	if (l==NULL)
    	{
    		lHeight=0;
    	}
    	else
    	{
    		lHeight= l->getHeight();
    	}
    	if (r==NULL)
    	{
    		rHeight=0;
    	}
    	else
    	{
    		rHeight=r->getHeight();
    	}
    	return lHeight-rHeight;
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    void AVL<Key, Elem, KEComp, EEComp>::updateHeight(BinNode<Elem>*& subroot)
    {
    	int factor, sonFactor;
    	bool isLeft;
    	BinNode<Elem> *son;
    	if (subroot==NULL)
    	{
    		return;
    	}

    	adjustHeight(subroot);

    	factor=getFactor(subroot);

    	isLeft=((AVLNodePtr<Elem>*)subroot)->getSide();
    	son=subroot->getSon(isLeft);
    	sonFactor=getFactor(son);
    	//first situation: the first 2/-2 we meet from bottom-up
    	if ((factor==2||factor==-2)&&subroot==root)
    	{
    		//a special case!!! as we search from bottom up
    		//we may wait to adjust until we reach its father
    		//the father happens to be root. But it is not a
    		//root adjustment!!!
    		if (sonFactor==1||sonFactor==-1||sonFactor==0)
    		{
    			adjust(subroot, true);
    		}
    		else
    		{
    			adjust(subroot, false);
    		}
    	}
    	else
    	{
    		if (sonFactor==2||sonFactor==-2)
    		{
    			adjust(subroot, false);
    		}
    	}
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    BinNode<Elem>* AVL<Key, Elem, KEComp, EEComp>::inserthelp(BinNode<Elem>* subroot,
    														  const Elem& val)
    {
    	if (subroot == NULL)            // Empty tree: create node
    	{
    		return (new AVLNodePtr<Elem>(val, NULL, NULL, 1));
    	}
    	if (EEComp::lt(val, subroot->val())) // Insert on left
    	{
    		((AVLNodePtr<Elem>*)subroot)->setSide(true);
    		subroot->setLeft(inserthelp(subroot->left(), val));
    		updateHeight(subroot);
    	}
    	else
    	{
    		((AVLNodePtr<Elem>*)subroot)->setSide(false);
    		subroot->setRight(inserthelp(subroot->right(), val));
    		updateHeight(subroot);
    	}
    	return subroot;  // Return subtree with node inserted
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    BinNode<Elem>* AVL<Key, Elem, KEComp, EEComp>::
    removehelp(BinNode<Elem>* subroot, const Key& K, BinNode<Elem>*& t)
    {
    	if (subroot == NULL)
    	{
    		return NULL; // Val is not in tree
    	}
    	else
    	{
    		if (KEComp::lt(K, subroot->val())) // Check left
    		{
    			((AVLNodePtr<Elem>*)subroot)->setSide(true);
    			subroot->setLeft(removehelp(subroot->left(), K, t));
    			//updateHeight(subroot);
    		}
    		else
    		{
    			if (KEComp::gt(K, subroot->val())) // Check right
    			{
    				((AVLNodePtr<Elem>*)subroot)->setSide(false);
    				subroot->setRight(removehelp(subroot->right(), K, t));
    				//updateHeight(subroot);
    			}
    			else
    			{                           // Found it: remove it
    				BinNode<Elem>* temp;
    				t = subroot;

    				if (subroot->left() == NULL)       // Only a right child
    				{
    					 subroot = subroot->right();      //  so point to right
    				}
    				else
    				{
    					if (subroot->right() == NULL) // Only a left child
    					{
    						subroot = subroot->left();       //  so point to left
    					}
    					else
    					{
    					// Both children are non-empty
    						subroot->setRight(deletemin(subroot->right(), temp));
    						Elem te = subroot->val();
    						subroot->setVal(temp->val());
    						temp->setVal(te);
    						t = temp;
    						((AVLNodePtr<Elem>*)subroot)->setSide(false);
    						//updateHeight(subroot);
    					}
    				}
    			}
    		}
    	}
    	updateHeight(subroot);
    	return subroot;
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    BinNode<Elem>* AVL<Key, Elem, KEComp, EEComp>::
    deletemin(BinNode<Elem>* subroot, BinNode<Elem>*& min)
    {
    	if (subroot->left() == NULL)
    	{ // Found min
    		min = subroot;
    		return subroot->right();
    	}
    	else
    	{                         // Continue left
    		((AVLNodePtr<Elem>*)subroot)->setSide(true);
    		subroot->setLeft(deletemin(subroot->left(), min));
    		updateHeight(subroot);
    		return subroot;
    	}
    }

    template <class Key, class Elem, class KEComp, class EEComp>
    void AVL<Key, Elem, KEComp, EEComp>::clearhelp(BinNode<Elem>* subroot)
    {
    	if (subroot == NULL)
    	{
    		return;
    	}
    	clearhelp(subroot->left());
    	clearhelp(subroot->right());
    	delete subroot;
    }

    file name: AVLTree.cpp  (main)

    #include <iostream>
    #include <time.h>
    #include "AVLTree.h"
    #include "Elem.h"

    using namespace std;

    int main()
    {
    	int num;
    	AVL<int, int, KEComp, EEComp> A;
    	//srand(time(0));
    	for (int i=0; i<20; i++)
    	{
    		cout<<"===================";
    		num=rand()%100+12;
    		cout<<"insert number "<<num<<endl;
    		A.insert(num);
    		A.print();
    	}
    	for (i=0; i<20; i++)
    	{
    		int temp;
    		cin>>num;
    		A.remove(num, temp);
    		cout<<"\nnow remove number"<<num<<endl;
    		A.print();
    	}

    	return 0;
    }

    Here is the result: Please note that there are

    single rotating while inserting number 90, 93, 107,

    double rotating while inserting number 36, 74,

    ===================insert number 53
    53
    ===================insert number 79
    53
      79
    ===================insert number 46
      46
    53
      79
    ===================insert number 12
        12
      46
    53
      79
    ===================insert number 81
        12
      46
    53
      79
        81
    ===================insert number 36
        12
      36
        46
    53
      79
        81
    ===================insert number 90
        12
      36
        46
    53
        79
      81
        90
    ===================insert number 70
        12
      36
        46
    53
          70
        79
      81
        90
    ===================insert number 74
        12
      36
        46
    53
          70
        74
          79
      81
        90
    ===================insert number 76
        12
      36
        46
    53
          70
        74
          76
      79
        81
          90

    input the number to remove: 79

    now remove number79
        12
      36
        46
    53
          70
        74
          76
      81
        90

    input the number to remove: 12

    now remove number12
      36
        46
    53
          70
        74
          76
      81
        90

    input the number to remove: 46

    now remove number46
        36
      53
        70
    74
        76
      81
        90

    input the number to remove: 81

    now remove number81
        36
      53
        70
    74
        76
      90

    input the number to remove: 76

    now remove number76
        36
      53
        70
    74
      90

    input the number to remove: 90

    now remove number90
      36
    53
        70
      74

    input the number to remove: 36

    now remove number36
      53
    70
      74

    input the number to remove: 70

    now remove number70
      53
    74

    input the number to remove: 74

    now remove number74
    53

    input the number to remove: 53

    now remove number53
    The BST is empty.
    Press any key to continue

    ¡¡