C++类实现AVL树

二叉查找树是个好东西，他让查找，插入，删除，这些常用操作变得高效，但是，他是存在问题的，那就是，在坏的输入序列下，树会退化成链表，这就很尴尬了，于是为了避免这种情况的发生，我们需要一种数据结构，可以自动对树进行调整，我们希望树尽量平衡，于是我们使用平衡因子作为指标，保持任意节点左右子树深度差不超过1，这就可以让树的深度很理想了（接近log2N)，如何对树进行调整呢？我们通过旋转来完成他。

#include <iostream>
#include <queue>

class AVLT{//小于向左，大于等于向右,元素类型需支持 <
private:
    typedef int Element;
    struct AVLTNode{
        Element data;
        AVLTNode *left;
        AVLTNode *right;
        int Deepth;
        AVLTNode();
        ~AVLTNode();
        bool isLeaf()const;
        void nodeInsect(const Element &e);
        void nodeDelete(const Element &e);
        const AVLTNode *Find(const Element &e)const;
        const Element& Min()const;
        const Element& Max()const;
        void nodeTraverseFrist(std::ostream& os = std::cout)const;
        void nodeTraverseMid(std::ostream& os = std::cout)const;
        void nodeTraverseLast(std::ostream& os = std::cout)const;
        void nodeTraverselevel(std::ostream& os = std::cout)const;
        const AVLTNode & operator =(const AVLTNode &T);
        void swapdata(AVLTNode&a, AVLTNode&b)const;
        void SingleLeftRotation();
        void SingleRightRotation();
        void DoubleRightLeftRotation();
        void DoubleLeftRightRotation();
        inline void renewDeepth();
        inline int max(int a, int b)const;
    };
    AVLTNode *node;
    int number_of_Node;
public:
    AVLT();
    AVLT(Element *arr, int size);
    AVLT(const AVLT &T);
    ~AVLT();
    bool empty()const;
    const Element& Max()const;
    const Element& Min()const;
    const AVLT &operator = (const AVLT&T);
    void TraverseFrist(std::ostream& os = std::cout)const;
    void TraverseLast(std::ostream& os = std::cout)const;
    void TraverseMid(std::ostream& os = std::cout)const;
    void Traverselevel(std::ostream& os = std::cout)const;
    void Insect(const AVLT::Element &e);
    bool Delete(const AVLT::Element &e);
    int Deepth();
    int Node();
    const AVLTNode* find(const Element &e)const;
};

AVLT::AVLTNode::AVLTNode():left(NULL), right(NULL), Deepth(){}
AVLT::AVLTNode::~AVLTNode(){
    delete left;
    delete right;
}
bool AVLT::AVLTNode::isLeaf()const{
    return left == NULL&&right == NULL;
}
void AVLT::AVLTNode::nodeInsect(const Element &e){
    if (Deepth == ){
        Deepth = ;
        data = e;
        return;
    }
    if (e<data){
        if (left == NULL){
            left = new AVLTNode;
        }
        left->nodeInsect(e);
        if (right){
            if (left->Deepth - right->Deepth > ){
                if (e<left->data){
                    this->SingleRightRotation();
                }
                else{
                    this->DoubleLeftRightRotation();
                }
            }
        }
        else{
            if (left->Deepth > ){
                if (e<left->data){
                    this->SingleRightRotation();
                }
                else{
                    this->DoubleLeftRightRotation();
                }
            }
        }
        renewDeepth();
    }
    else{
        if (right == NULL){
            right = new AVLTNode;
        }
        right->nodeInsect(e);
        if (left){
            if (right->Deepth - left->Deepth > ){
                if (e<right->data){
                    this->DoubleRightLeftRotation();
                }
                else{
                    this->SingleLeftRotation();
                }
            }
        }
        else{
            if (right->Deepth > ){
                if (e<right->data){
                    this->DoubleRightLeftRotation();
                }
                else{
                    this->SingleLeftRotation();
                }
            }
        }
        renewDeepth();
    }
    return;
}
void AVLT::AVLTNode::nodeDelete(const Element &e){
    if (e<data){
        if (left){
            if (left->isLeaf()){
                delete left;
                left = NULL;
            }
            else
                left->nodeDelete(e);
            {
                int rd = right ? right->Deepth : ;
                int ld = left ? left->Deepth : ;
                if (rd - ld >){
                    if (right->left){
                        DoubleRightLeftRotation();
                    }
                    else{
                        SingleLeftRotation();
                    }
                }
                renewDeepth();
            }
        }
        return;
    }
    else{
        if (data<e){
            if (right){
                if (right->isLeaf()){
                    delete right;
                    right = NULL;
                }
                else
                    right->nodeDelete(e);
                {
                    int rd = right ? right->Deepth : ;
                    int ld = left ? left->Deepth : ;
                    if (ld - rd >){
                        if (left->right){
                            DoubleLeftRightRotation();
                        }
                        else{
                            SingleRightRotation();
                        }
                    }
                    renewDeepth();
                }
                return;
            }
        }
        else{
            if (left == NULL&&right){
                AVLTNode *t = right;
                if (left)
                    delete left;
                left = t->left;
                right = t->right;
                data = t->data;
                Deepth -= ;
                t->left = t->right = NULL;
                delete t;
                return;
            }
            else{
                if (right == NULL && left){
                    AVLTNode *t = left;
                    if (right)
                        delete right;
                    right = t->right;
                    left = t->left;
                    data = t->data;
                    Deepth -= ;
                    t->left = t->right = NULL;
                    delete t;
                    return;
                }
                else{

                    AVLTNode *t = left;
                    while (t->right)
                        t = t->right;
                    Element te = t->data;
                    if (left->isLeaf()){
                        delete left;
                        left = NULL;
                    }
                    else
                        left->nodeDelete(te);
                    data = te;
                    return;
                }
            }
        }
    }
    std::cout << "return false" << std::endl;
    return;
}
AVLT::AVLTNode const *AVLT::AVLTNode::Find(const Element &e)const{
    if (e<data){
        if (left)
            return left->Find(e);
    }
    else{
        if (data<e){
            if (right)
                return right->Find(e);
        }
        else{
            return this;
        }
    }
    return NULL;
}
const AVLT::Element& AVLT::AVLTNode::Max()const{
    const AVLTNode *t = this;
    while (t->right)
        t = t->right;
    return t->data;
}
const AVLT::Element& AVLT::AVLTNode::Min()const{
    const AVLTNode *t = this;
    while (t->left)
        t = t->left;
    return t->data;
}
void AVLT::AVLTNode::nodeTraverseFrist(std::ostream& os)const{
    os << this->data << ' ';
    if (left)
        this->left->nodeTraverseFrist();
    if (right)
        this->right->nodeTraverseFrist();
}
void AVLT::AVLTNode::nodeTraverseMid(std::ostream& os )const{
    if (left)
        this->left->nodeTraverseMid();
    os << this->data << ' ';
    if (right)
        this->right->nodeTraverseMid();
}
void AVLT::AVLTNode::nodeTraverseLast(std::ostream& os )const{
    if (left)
        this->left->nodeTraverseLast();
    if (right)
        this->right->nodeTraverseLast();
    os << this->data << ' ';
}
void AVLT::AVLTNode::nodeTraverselevel(std::ostream& os)const{
    std::queue<AVLTNode> q;
    q.push(*this);
    while (!q.empty()){
        AVLTNode p = q.front();
        q.pop();
        os << p.data << ' ';
        if (p.left){
            q.push(*(p.left));
        }
        if (p.right){
            q.push(*(p.right));
        }
    }
}
const AVLT::AVLTNode & AVLT::AVLTNode::operator =(const AVLTNode &T){
    if (Deepth == ){
        Deepth = ;
        data = T.data;
    }
    if (T.left){
        if (left == NULL){
            left = new AVLTNode;
        }
        *left = *T.left;
        if (right){
            Deepth = max(left->Deepth, right->Deepth) + ;
        }
        else{
            Deepth = left->Deepth + ;
        }
    }
    if (T.right){
        if (right == NULL){
            right = new AVLTNode;
        }
        *right = *T.right;
        if (left){
            Deepth = max(left->Deepth, right->Deepth) + ;
        }
        else{
            Deepth = right->Deepth + ;
        }
    }
    return *this;
}
void AVLT::AVLTNode::swapdata(AVLTNode&a, AVLTNode&b)const{
    Element t;
    t = a.data;
    a.data = b.data;
    b.data = t;
}
void AVLT::AVLTNode::SingleLeftRotation(){
    AVLTNode *a = right;
    right = a->right;
    a->right = a->left;
    a->left = left;
    left = a;
    if (a->right&&a->left){
        a->Deepth = max(a->left->Deepth, a->right->Deepth) + ;
    }
    else
        if (a->left){
            a->Deepth = a->left->Deepth + ;
        }
        else{
            a->Deepth = ;
        }
        swapdata(*this, *a);
}
void AVLT::AVLTNode::SingleRightRotation(){
    AVLTNode *a = left;
    left = a->left;
    a->left = a->right;
    a->right = right;
    right = a;
    if (a->right&&a->left){
        a->Deepth = max(a->left->Deepth, a->right->Deepth) + ;
    }
    else
        if (a->left){
            a->Deepth = a->left->Deepth + ;
        }
        else{
            a->Deepth = ;
        }
        swapdata(*this, *a);
}
void AVLT::AVLTNode::DoubleLeftRightRotation(){
    this->left->SingleLeftRotation();
    this->SingleRightRotation();
}
void AVLT::AVLTNode::DoubleRightLeftRotation(){
    this->right->SingleRightRotation();
    this->SingleLeftRotation();
}
inline void AVLT::AVLTNode::renewDeepth(){
    Deepth = max(left ? left->Deepth : , right ? right->Deepth : ) + ;
}
inline int AVLT::AVLTNode::max(int a, int b)const{
    return a > b ? a : b;
}

AVLT::AVLT():node(NULL), number_of_Node(){
}
AVLT::AVLT(Element *arr, int size) : node(NULL), number_of_Node(){
    node = new AVLTNode;
    for (int i = ; i < size; i++)
        this->Insect(arr[i]);
}
AVLT::AVLT(const AVLT &T) : node(NULL), number_of_Node(){
    *this = T;
}
AVLT::~AVLT(){
    delete node;
}
bool AVLT::empty()const{
    return node == NULL;
}
const AVLT::Element& AVLT::Max()const{
    return node->Max();
}
const AVLT::Element& AVLT::Min()const {
    return node->Min();
}
const AVLT &AVLT::operator = (const AVLT&T){
    delete node;
    node = new AVLTNode;
    number_of_Node = T.number_of_Node;
    *node = *T.node;
    return *this;
}
void AVLT::TraverseFrist(std::ostream& os )const{
    if (node)
        node->nodeTraverseFrist(os);
}
void AVLT::TraverseLast(std::ostream& os )const{
    if (node)
        node->nodeTraverseLast(os);
}
void AVLT::TraverseMid(std::ostream& os )const{
    if (node)
        node->nodeTraverseMid(os);
}
void AVLT::Traverselevel(std::ostream& os )const{
    if (node)
        node->nodeTraverselevel(os);
}
void AVLT::Insect(const AVLT::Element &e){
    number_of_Node++;
    if (node == NULL){
        node = new AVLTNode;
    }
    node->nodeInsect(e);
}
bool AVLT::Delete(const AVLT::Element &e){
    if (node){
        if (node->isLeaf()){
            delete node;
            node = NULL;
        }
        else
            node->nodeDelete(e);
        number_of_Node--;
        return true;
    }
    else{
        return false;
    }
}
int AVLT::Deepth(){
    return node->Deepth;
}
int AVLT::Node(){
    return number_of_Node;
}
const AVLT::AVLTNode*AVLT::find(const AVLT::Element &e)const {
    if (node)
        return node->Find(e);
    else
        return NULL;;
}
std::ostream& operator << (std::ostream& os, const AVLT& T){
    T.TraverseFrist(os);
    return os;
}
const AVLT& find();

这样看上去，AVL树似乎完胜普通的二叉查找树，但是现实是，坏的序列往往很少出现，很多时候，普通二叉树因为不需要判断平衡因子，旋转这些操作，效率反而更高，但是他一旦对上坏的序列就无计可施了，想能防住坏的序列，但又不想每次都旋转，于是就有伸展树这种数据结构，他只在坏的序列出现时旋转，但是只有“坏到一定

程度才旋转”，于是会出现比较坏但坏得不彻底的尴尬情况，伸展树无法保证每次操作都是O（log2N)，但是能保证M次操作时间复杂度为O（Mlog2N）。