【愚公系列】2021年11月 C#版 数据结构与算法解析(AVL树)

AVL树定义：在计算机科学中，AVL树是最先发明的自平衡二叉查找树。在AVL树中任何节点的两个子树的高度最大差别为1，所以它也被称为高度平衡树。增加和删除可能需要通过一次或多次树旋转来重新平衡这个树。AVL树得名于它的发明者G. M. Adelson-Velsky和E. M. Landis，他们在1962年的论文《An algorithm for the organization of information》中发表了它。

平衡二叉树是带有平衡条件的二叉查找树，指的是空树或者任一结点左、右高度差的绝对值不超过1的二叉树.适合用于插入删除次数比较少，但查找多的情况。

比如:

实现的难点在于，二叉树的平衡旋转

分为四种旋转，RR、LL、LR、RL旋转

RR旋转

麻烦结点在发现者右子树的右边，所以叫RR插入，需要RR旋转

LL旋转

麻烦结点在发现者左子树的左边，所以叫LL插入，需要LL旋转

LR旋转

RL旋转

C#实现源码

public class Bit : IComparable<Bit>
{
    public int Value;

    public Bit(int value) => Value = value;

    public int CompareTo(Bit other) => Value - other.Value;
}
public class AvlTreeNote<TKey> where TKey : IComparable<TKey>
{
    public TKey Key;
    public int Height;
    public AvlTreeNote<TKey> LChild;
    public AvlTreeNote<TKey> RChild;

    public AvlTreeNote(TKey key, AvlTreeNote<TKey> lChild, AvlTreeNote<TKey> rChild)
    {
        Key = key;
        LChild = lChild;
        RChild = rChild;
    }
}
public class AvlTree<TKey> where TKey : IComparable<TKey>
{
    public AvlTreeNote<TKey> Root;          // 根节点

    private bool _isBalance;                // 标志是否平衡过二叉树

    /// <summary>
    /// 插入节点
    /// </summary>
    /// <param name="key"></param>
    /// <returns></returns>
    public AvlTreeNote<TKey> Insert(TKey key) => Root = Insert(key, Root);

    /// <summary>
    /// 插入到指定节点下
    /// </summary>
    /// <param name="key"></param>
    /// <param name="node"></param>
    /// <returns></returns>
    private AvlTreeNote<TKey> Insert(TKey key, AvlTreeNote<TKey> node)
    {
        if (node == null)
        {
            node = new AvlTreeNote<TKey>(key, null, null);
        }
        else
        {
            // 如果树里已经存在该节点，直接返回为null

            if (key.CompareTo(node.Key) == 0) return null;

            if (key.CompareTo(node.Key) < 0)
            {
                // 应该在左树进行搜索插入

                node.LChild = Insert(key, node.LChild);

                if (node.LChild == null) return node;

                switch (node.Height)
                {
                    case 1:
                        return LeftBalance(node);
                    case 0:
                        node.Height = _isBalance ? 0 : 1;
                        break;
                    case -1:
                        node.Height = 0;
                        break;
                }
            }
            else
            {
                // 应该在右树进行搜索插入

                node.RChild = Insert(key, node.RChild);

                if (node.RChild == null) return node;

                switch (node.Height)
                {
                    case 1:
                        node.Height = 0;
                        break;
                    case 0:
                        node.Height = _isBalance ? 0 : -1;
                        break;
                    case -1:
                        return RightBalance(node);
                }
            }
        }

        _isBalance = false;

        return node;
    }

    /// <summary>
    /// 左树平衡处理
    /// </summary>
    /// <param name="node"></param>
    private AvlTreeNote<TKey> LeftBalance(AvlTreeNote<TKey> node)
    {
        if (_isBalance) return node;

        var leftNode = node.LChild;

        switch (leftNode.Height)
        {
            case 1:

                node.Height = leftNode.Height = 0;

                node = R_Rotate(node);

                break;

            case -1:

                node.Height = leftNode.Height = 0;

                node.LChild = L_Rotate(leftNode);

                node = R_Rotate(node);

                break;
        }

        return node;
    }

    private AvlTreeNote<TKey> RightBalance(AvlTreeNote<TKey> node)
    {
        if (_isBalance) return node;

        var rightNode = node.RChild;

        switch (rightNode.Height)
        {
            case -1:

                node.Height = rightNode.Height = 0;

                node = L_Rotate(node);

                break;

            case 1:

                node.Height = rightNode.Height = 0;

                node.RChild = R_Rotate(rightNode);

                node = L_Rotate(node);

                break;
        }

        return node;
    }

    /// <summary>
    /// 右旋操作
    /// </summary>
    /// <param name="node"></param>
    private AvlTreeNote<TKey> R_Rotate(AvlTreeNote<TKey> node)
    {
        var temp = node.LChild;

        node.LChild = temp.RChild;

        temp.RChild = node;

        _isBalance = true;

        return temp;
    }

    /// <summary>
    /// 左旋操作
    /// </summary>
    /// <param name="node"></param>
    private AvlTreeNote<TKey> L_Rotate(AvlTreeNote<TKey> node)
    {
        var temp = node.RChild;

        node.RChild = temp.LChild;

        temp.LChild = node;

        _isBalance = true;

        return temp;
    }

    /// <summary>
    /// 查找二叉树
    /// </summary>
    /// <param name="key"></param>
    /// <returns></returns>
    public AvlTreeNote<TKey> Find(TKey key) => Find(key, Root);

    /// <summary>
    /// 查找二叉树
    /// </summary>
    /// <param name="key"></param>
    /// <param name="node"></param>
    /// <returns></returns>
    public AvlTreeNote<TKey> Find(TKey key, AvlTreeNote<TKey> node)
    {
        if (node == null) return null;

        if (key.CompareTo(node.Key) < 0)
        {
            node = Find(key, node.LChild);
        }
        else if (key.CompareTo(node.Key) > 0)
        {
            node = Find(key, node.RChild);
        }

        return node;
    }

    /// <summary>
    /// 用于删除该节点移动节点
    /// </summary>
    /// <param name="node"></param>
    /// <param name="findNode"></param>
    /// <returns></returns>
    private AvlTreeNote<TKey> Move(AvlTreeNote<TKey> node, AvlTreeNote<TKey> findNode)
    {
        AvlTreeNote<TKey> moveNode;

        if (findNode != null)
        {
            if (findNode.RChild != null)
            {
                moveNode = findNode.RChild;

                findNode.RChild = null;
            }
            else
            {
                findNode.LChild = null;

                moveNode = findNode;
            }

            if (node.LChild != moveNode) moveNode.LChild = node.LChild;

            if (node.RChild != moveNode) moveNode.RChild = node.RChild;
        }
        else
        {
            moveNode = null;
        }

        node.LChild = null;

        node.RChild = null;

        node.Key = default(TKey);

        node.Height = 0;

        return moveNode;
    }

    /// <summary>
    /// 删除节点
    /// </summary>
    /// <param name="key"></param>
    public void Remove(TKey key) => Root = Remove(key, Root);

    private AvlTreeNote<TKey> Remove(TKey key, AvlTreeNote<TKey> node)
    {
        if (node == null) return null;

        if (key.CompareTo(node.Key) < 0)
        {
            if (node.LChild == null) return node;

            node.LChild = Remove(key, node.LChild);

            switch (node.Height)
            {
                case 1:
                    node.Height = 0;
                    break;
                case 0:
                    node.Height = -1;
                    break;
                case -1:
                    // 要进行旋转
                    node.Height = 0;
                    return node.LChild == null ? RightBalance(node) : LeftBalance(node);
            }
        }
        else if (key.CompareTo(node.Key) > 0)
        {
            if (node.RChild == null) return node;

            node.RChild = Remove(key, node.RChild);

            switch (node.Height)
            {
                case 1:
                    // 要进行旋转
                    node.Height = 0;
                    return node.RChild == null ? LeftBalance(node) : RightBalance(node);
                    break;
                case 0:
                    node.Height = 1;
                    break;
                case -1:
                    node.Height = 0;
                    break;
            }
        }
        else if (key.CompareTo(node.Key) == 0)
        {
            var findNode = Remove(key, node.LChild);

            node = Move(node, findNode);
        }

        _isBalance = false;

        return node;
    }
}      

C/C++实现

#include <stack>//栈
#include <queue>
#include <iostream>
#include <initializer_list>
using namespace std;
template <typename Comparable>
class AVLTree
{
private:
    static const int ALLOWED_IMBLANCE = 1;
    struct AVLNode
    {
        Comparable element;
        AVLNode * left;
        AVLNode * right;
        int height;

        AVLNode(const Comparable & theElement, AVLNode *lt, AVLNode *rt,int h = 0)
            :element(theElement), left(lt), right(rt),height(h) {}
        AVLNode(Comparable && theElement, AVLNode *lt, AVLNode *rt, int h = 0)
            :element(std::move(theElement)), left(lt), right(rt),height(h) {}
    };
    AVLNode * root;
    void Insert(const Comparable &x, AVLNode * & t);
    void Insert(Comparable && x, AVLNode *&t);
    void Insert(initializer_list<Comparable> &d, AVLNode *& t);
    void Remove(const Comparable &x, AVLNode *&t);
    AVLNode * findMin(AVLNode *t)const;
    AVLNode * findMax(AVLNode *t)const;
    bool contains(const Comparable &x, AVLNode *t) const;
    void makeEmpty(AVLNode * &t);
    void PrintTree(AVLNode *t) const;
    AVLNode* clone(AVLNode *t)const
    {
        if (t == nullptr)
            return nullptr;
        return new AVLNode(t->element, clone(t->left), clone(t->right));
    }
    void rotateWithLeftChild(AVLNode *& k2);
    void rotateWithRightChild(AVLNode *& k2);
    void doubleWithLeftChild(AVLNode *&k3);
    void doubleWithRightChild(AVLNode *&k3);
    void balance(AVLNode*& t);
public:
    AVLTree() :root(nullptr) {}
    AVLTree(const AVLTree & rhs) :root(nullptr) { root = clone(rhs.root); }
    AVLTree(AVLTree && rhs) :root(nullptr) {    root = rhs.root;rhs = nullptr;}
    ~AVLTree() { makeEmpty(); }
    const Comparable & findMin() const { return findMin(root)->element; }
    const Comparable & findMax() const { findMax(root)->element; }
    bool contains(const Comparable & x) const { return contains(x, root); }
    bool isEmpty() const { return root == nullptr; }
    int height(AVLNode *t) const { return t == nullptr ? -1 : t->height; }
    void PrintTree()const { PrintTree(root); }
    void makeEmpty() { makeEmpty(root); }
    void Insert(const Comparable &x) { Insert(x,root); }
    void Insert(Comparable && x) { Insert(x,root); }
    void Insert(initializer_list<Comparable>d) { Insert(d, root); }
    void Remove(const Comparable &x) { Remove(x, root); }
    AVLTree & operator=(const AVLTree & rhs)
    {
        if (this != &rhs)
        {
            makeEmpty();
            root = clone(rhs.root);
        }
        return *this;
    }
    AVLTree & operator=(AVLTree && rhs)
    {
        if (this != &rhs)
        {
            makeEmpty();
            root = rhs.root;
            rhs.root = nullptr;
        }
        return *this;
    }
    friend int Max(int a, int b);

};
int Max(int a, int b)
{
    return (a > b) ? a : b;
}
template<typename Comparable>
void AVLTree<Comparable>::Insert(const Comparable & x, AVLNode *& t)
{
    if (t == nullptr)
    {
        t = new AVLNode(x, nullptr, nullptr);
    }
    else if (x < t->element)
        Insert(x, t->left);
    else if (t->element < x)
        Insert(x, t->right);

    balance(t);

}

template<typename Comparable>
void AVLTree<Comparable>::Insert(Comparable && x, AVLNode *& t)
{

    if (t == nullptr)
    {
        t = new AVLNode(std::move(x), nullptr, nullptr);
    }
    else if (x < t->element)
        Insert(std::move(x), t->left);
    else if (t->element < x)
        Insert(std::move(x), t->right);
    balance(t);
}

template<typename Comparable>
void AVLTree<Comparable>::Insert(initializer_list<Comparable> &d, AVLNode *& t)
{
    for (auto p = d.begin(); p != d.end(); p++)
    {
        Insert(*p, t);
    }
}
template<typename Comparable>
void AVLTree<Comparable>::Remove(const Comparable & x, AVLNode *& t)
{
    if (t == nullptr) //没找到相应的项什么都不做
        return;
    if (x < t->element)
        Remove(x, t->left);
    else if (x > t->element)
        Remove(x, t->right);
    else if (t->left != nullptr && t->right != nullptr)//找到了 但是有两个儿子
    {//取右子树的最小元素替代 或者左子树的最大元素，好处是右子树的最小元素一定在右子树的最左边，左子树的最大元素一定在左子树的最右边
        t->element = findMin(t->right)->element;//在右子树中找到最小的元素填充删除结点
        Remove(t->element, t->right);//在删除节点的右子树中删除最小元素.
    }
    else //有一个儿子 或者没有
    {
        AVLNode * oldNode = t;
        t = (t->left != nullptr) ? t->left : t->right; //如果有儿子，就让儿子接上，如果没有那t就设置为nullptr
        delete oldNode;
    }
    balance(t);
}

template<typename Comparable>
typename AVLTree<Comparable>::AVLNode *
AVLTree<Comparable>::findMin(AVLNode * t) const
{
    //递归版本
    //if (t == nullptr)  
    //  return nullptr;
    //if (t->left == nullptr)
    //  return t;
    //return findMin(t->left);

    //非递归版本
    if (t != nullptr)
        while (t->left != nullptr)
            t = t->right;
    return t;

}

template<typename Comparable>
typename AVLTree<Comparable>::AVLNode *
AVLTree<Comparable>::findMax(AVLNode * t)const
{
    //递归版本
    /*if (t == nullptr)
        return;
    if (t->right == nullptr)
        return t;
    return findMax(t->right);*/
    if (t != nullptr)
        while (t->right != nullptr)
            t = t->right;
    return t;


}

template<typename Comparable>
bool AVLTree<Comparable>::contains(const Comparable & x, AVLNode * t) const
{
    //递归版本
    /*if (t == nullptr)
        return false;
    else if (x < t->element)
        return contains(x, t->left);
    else if (x > t->element)
        return contains(x, t->right);
    else
        retu true;*/

        //非递归版本
    while (t != nullptr)
    {
        if (x < t->element)
            t = t->left;
        else if (x > t->element)
            t = t->right;
        else
            return true;
    }
    return false;
}

template<typename Comparable>
void AVLTree<Comparable>::makeEmpty(AVLNode *& t)
{

    if (t != nullptr)
    {
        makeEmpty(t->left);
        makeEmpty(t->right);
        delete t;
    }
    t = nullptr;

}

template<typename Comparable>
void AVLTree<Comparable>::PrintTree(AVLNode * t) const
{
    //递归先序遍历
    if (t != nullptr)
    {
        std::cout << t->element << " "; //先序遍历
        PrintTree(t->left);
        PrintTree(t->right);
    }
    //非递归的先序遍历， 使用栈
    //AVLNode * temp = t;
    //stack<AVLNode*>s;
    //while (temp || !s.empty())
    //{
    //  while (temp) //一直向左将沿途结点压入栈
    //  {
    //      s.push(temp);
    //      temp = temp->left;
    //  }
    //  if (!s.empty())
    //  {
    //      temp = s.top();
    //      s.pop();
    //      std::cout << temp->element << " "; //先序遍历
    //      temp = temp->right;
    //  }
    //}
    //层序遍历，使用队列
    //AVLNode * temp;
    //queue<AVLNode*>q;
    //if (t == nullptr)
    //  return;
    //q.push(t);
    //while (!q.empty())
    //{
    //  temp = q.front();
    //  q.pop();
    //  std::cout << temp->element << " ";
    //  if (temp->left)
    //      q.push(temp->left);
    //  if (temp->right)
    //      q.push(temp->right);
    //}


}

template<typename Comparable>
void AVLTree<Comparable>::rotateWithLeftChild(AVLNode *& k2)//左旋
{
    AVLNode *k1 = k2->left;
    k2->left = k1->right;
    k1->right = k2;
    k2->height = Max(height(k2->left), height(k2->right)) + 1;
    k1->height = Max(height(k1->left), k2->height) + 1;
    k2 = k1;//把所有的设置都变为改变后的设置
}

template<typename Comparable>
void AVLTree<Comparable>::rotateWithRightChild(AVLNode *& k2)//右旋
{
    AVLNode *k1 = k2->right;
    k2->right = k1->left;
    k1->left = k2;
    k2->height = Max(height(k2->right), height(k2->left)) + 1;
    k1->height = Max(height(k1->right), k2->height) + 1;
    k2 = k1;

}

template<typename Comparable>
void AVLTree<Comparable>::doubleWithLeftChild(AVLNode *& k3)//左右旋转
{
    rotateWithRightChild(k3->left);
    rotateWithLeftChild(k3);
}

template<typename Comparable>
void AVLTree<Comparable>::doubleWithRightChild(AVLNode *& k3)//右左旋转
{
    rotateWithLeftChild(k3->right);
    rotateWithRightChild(k3);
}

template<typename Comparable>
void AVLTree<Comparable>::balance(AVLNode *& t)
{
    if (t == nullptr)
    {
        return;
    }
    if (height(t->left) - height(t->right) > ALLOWED_IMBLANCE)
        if (height(t->left->left) >= height(t->left->right))
            rotateWithLeftChild(t);
        else
            doubleWithLeftChild(t);
    else if(height(t->right) - height(t->left) > ALLOWED_IMBLANCE)
        if (height(t->right->right) >= height(t->right->left))
            rotateWithRightChild(t);
        else
            doubleWithRightChild(t);
    t->height = max(height(t->left), height(t->right)) + 1;
}