算法与数据结构——AVL树（平衡二叉树）

定义

typedef struct AVLNode* Position;
typedef Position AVLTree; /* AVL树类型 */
struct AVLNode {
    ElementType Data; /* 结点数据 */
    AVLTree Left;     /* 指向左子树 */
    AVLTree Right;    /* 指向右子树 */
    int Height;       /* 树高 */
};

前期准备函数

int Max(int a, int b)
{
    return a > b ? a : b;
}
int GetHeight(Position T)
{
    int hr, hl;
    if (!T)
    {
        return 0;
    }
    hl = GetHeight(T->Left);
    hr = GetHeight(T->Right);
    return (hl > hr) ? ++hl : ++hr;
}

实现功能的主要函数

调整函数

Position SingleRoateWithLeft(Position A)
{
    Position B;
    B = A->Left;
    A->Left = B->Right;
    B->Right = A;
    A->Height = Max(GetHeight(A->Left), GetHeight(A->Right)) + 1;
    B->Height = Max(GetHeight(B->Left), GetHeight(B->Right)) + 1;
    return B;//new node
}
//右单旋
Position SingleRoateWithRight(Position A)
{
    Position B;
    B = A->Right;
    A->Right = B->Left;
    B->Left = A;
    A->Height = Max(GetHeight(A->Left), GetHeight(A->Right)) + 1;
    B->Height = Max(GetHeight(B->Left), GetHeight(B->Right)) + 1;
    return B;
}
//右左双旋
Position DoubleRoateWithLeft(Position A)
{
    A->Left = SingleRoateWithRight(A->Left);
    return SingleRoateWithLeft(A);
}
//左右双旋
Position DoubleRoateWithRight(Position A)
{
    A->Right = SingleRoateWithLeft(A->Right);
    return SingleRoateWithRight(A);
}

插入函数（兼具建树功能）

AVLTree Insert(ElementType X, AVLTree T)
{
    if (T == NULL)
    {
        T = new AVLNode;
        T->Data = X;
        T->Left = T->Right = NULL;
        T->Height = 0;
    }
    else if (X < T->Data)
    {
        T->Left = Insert(X, T->Left);
        if (GetHeight(T->Left) - GetHeight(T->Right) == 2)
        {
            if (X < T->Left->Data)
            {
                T = SingleRoateWithLeft(T);
            }
            else
            {
                T = DoubleRoateWithLeft(T);
            }
        }
    }
    else
    {
        T->Right = Insert(X, T->Right);
        if (GetHeight(T->Right) - GetHeight(T->Left) == 2)
        {
            if (X > T->Right->Data)
            {
                T = SingleRoateWithRight(T);
            }
            else
            {
                T = DoubleRoateWithRight(T);
            }
        }

    }
    T->Height = Max(GetHeight(T->Left), GetHeight(T->Right)) + 1;
    return T;
}

删除函数

删除函数和插入函数很像，删除的情况要考虑好。图解参考

AVLTree Delete(ElementType X, AVLTree T)
{
    if (T == NULL)
    {
        return NULL;
    }
    if (X < T->Data)
    {
        T->Left = Delete(X, T->Left);
    }
    else if (X > T->Data)
    {
        T->Right = Delete(X, T->Right);
    }
    else
    {
        if (T->Left)
        {
           AVLTree tmp= FindMax(T->Left);
           T->Data = tmp->Data;
           T->Left = Delete(tmp->Data, T->Left);
        }
        else if (T->Right)
        {
            AVLTree tmp = FindMin(T->Right);
            T->Data = tmp->Data;
            T->Right = Delete(tmp->Data, T->Right);
        }
        else
        {
            free(T);
            return NULL;
        }
    }
    if (GetHeight(T->Left) - GetHeight(T->Right) == 2)
    {
        if (GetHeight(T->Left->Left)>GetHeight(T->Left->Right))
        {
            T = SingleRoateWithLeft(T);
        }
        else
        {
            T = DoubleRoateWithLeft(T);
        }
    }
    else if (GetHeight(T->Right) - GetHeight(T->Left) == 2)
    {
        if (GetHeight(T->Right->Right) > GetHeight(T->Right->Left))
        {
            T = SingleRoateWithRight(T);
        }
        else
        {
            T = DoubleRoateWithRight(T);
        }
    }
    
    T->Height = Max(GetHeight(T->Left), GetHeight(T->Right)) + 1;
    return T;
}

其他函数

VLTree MakeEmpty(AVLTree T)
{
    if (T != NULL)
    {
        MakeEmpty(T->Left);
        MakeEmpty(T->Right);
        free(T);
    }
    return NULL;
}
Position Find(ElementType X, AVLTree T)
{
    if (T == NULL)
    {
        return NULL;
    }
    if (X < T->Data)
    {
        return Find(X,T->Left);
    }
    else if(X>T->Data)
    {
        return Find(X, T->Right);
    }
    else
    {
        return T;
    }
}
Position FindMin(AVLTree T)
{
    if (T == NULL)
    {
        return NULL;
    }
    else if(T->Left==NULL)
    {
        return T;
    }
    else
    {
        return FindMin(T->Left);
    }

}
Position FindMax(AVLTree T)
{
    if (T == NULL)
    {
        return NULL;
    }
    else if(T->Right==NULL)
    {
        return T;
    }
    else
    {
        return FindMax(T->Right);
    }
}

深度和高度

void showHeight(int a,Position T)
{

     if (T == NULL)
     {
         return;
     }
     showHeight(a,T->Left);
     cout << T->Height << ' ';
     showHeight(a,T->Right);
}
  void showdeep(int depth,Position T)
  {
     if (T == NULL) return;
      if (T)
      {
          
          showdeep(depth + 1, T->Left);
          cout << depth+1<<' ';
          showdeep(depth + 1, T->Right);
      }
  }