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
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