AVL树的模板实现（增加了remove的方法）

 http://www.staroceans.com/AVLTree-remove.htm

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 classtemplate <class Elem> class BinNode {public:// Return the node's elementvirtual Elem& val() = 0;// Set the node's elementvirtual void setVal(const Elem&) = 0;// Return the node's left childvirtual BinNode* left() const = 0;// Set the node's left childvirtual void setLeft(BinNode*) = 0;// Return the node's right childvirtual BinNode* right() const = 0;// Set the node's right childvirtual void setRight(BinNode*) = 0;// Return true iff the node is a leafvirtual bool isLeaf() = 0;//my personal preferencevirtual BinNode<Elem>* getSon(bool isLeft)const=0; //my personal preferencevirtual void setSon(BinNode<Elem>* son, bool isLeft)=0;};// Binary tree node classtemplate <class Elem>class BinNodePtr : public BinNode<Elem> {private:Elem it;                     // The node's valueBinNodePtr* lc;              // Pointer to left childBinNodePtr* rc;              // Pointer to right childpublic:// Two constructors -- with and without initial valuesBinNodePtr() { lc = rc = NULL; }BinNodePtr(Elem e, BinNodePtr* l =NULL, BinNodePtr* r =NULL){ it = e; lc = l; rc = r; }~BinNodePtr() {}             // DestructorElem& 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 valueAVLNodePtr* lc;              // Pointer to left childAVLNodePtr* rc;              // Pointer to right childint height;bool inLeft;public: // Two constructors -- with and without initial valuesAVLNodePtr() { 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() {}             // DestructorElem& 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 ADTtemplate <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 systemclass 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 intemplate <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 referencetemplate<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 dosonFactor=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 signleft2right=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 checkisLeft=((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 isRootsubroot=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-upif ((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 itBinNode<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-emptysubroot->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 minmin = 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 5353===================insert number 7953  79===================insert number 46  4653  79===================insert number 12    12  4653  79===================insert number 81    12  4653  79    81===================insert number 36    12  36    4653  79    81===================insert number 90    12  36    4653    79  81    90===================insert number 70    12  36    4653      70    79  81    90===================insert number 74    12  36    4653      70    74      79  81    90===================insert number 76    12  36    4653      70    74      76  79    81      90input the number to remove: 79now remove number79    12  36    4653      70    74      76  81    90input the number to remove: 12now remove number12  36    4653      70    74      76  81    90input the number to remove: 46now remove number46    36  53    7074    76  81    90input the number to remove: 81now remove number81    36  53    7074    76  90input the number to remove: 76now remove number76    36  53    7074  90input the number to remove: 90now remove number90  3653    70  74input the number to remove: 36now remove number36  5370  74input the number to remove: 70now remove number70  5374input the number to remove: 74now remove number7453input the number to remove: 53now remove number53The BST is empty.Press any key to continue