二叉查找树的基本操作

二叉查找树：对于树中的每个节点X，它的左子树所有关键字小于X的关键字，而右子树的所有关键字大于X的关键字。

二叉查找树的平均深度是O(logN)。

二叉查找树的删除操作：

1. 如果节点是一片树叶，可以立即删除。
2. 如果有一个儿子，调节父节点指针绕过该节点后被删除。
3. 如果有两个儿子，用右子树的最小数据代替该节点的数据，并递归地删除那个节点（现在它是空的）。因为右子树的最小节点不可能有左儿子，所以第二次删除更容易。

#ifndef _Tree_H

struct TreeNode;
typedef struct TreeNode *Position;
typedef struct TreeNode *SearchTree;

SearchTree MakeEmpty(SearchTree T);
Position Find(ElementType X, SearchTree T);
Position FindMin(SearchTree T);
Position FindMax(SearchTree T);
SearchTree Insert(ElementType X, SearchTree T);
SearcgTree Delete(ElementType X, SearchTree T);
ElementType Retrieve(Position P);

#endif

struct TreeNode
{
    ElementType Element;
    SearchTree Left;
    SearchTree Right;
}


SearchTree MakeEmpty(SearchTree T)
{
    if (T != NULL)
    {
        MakeEmpty(T->Left);
        MakeEmpty(T->Right);
        free(T);
    }
    return NULL;
}


Position Find(ElementType X, SearchTree T)
{
    if (T == NULL)
        return NULL;
    if (X < T->Element)
        return Find(X, T->Left);
    else if (X > T->Element)
        return Find(X, T->Right);
    else
        return T;
}


Position FindMin(SearchTree T)
{
    if (T == NULL)
        return NULL;
    else if (T->Left == NULL)
        return T;
    else
        return FindMin(T->Left);
}

Position FindMax(SearchTree T)
{
    if (T != NULL)
        while (T->Right != NULL)
            T = T->Right;
    return T;
}


SearchTree Insert(ElementType X, SearchTree T)
{
    if (T == NULL)
    {
        T = malloc(sizeof(struct TreeNode));
        if (T = NULL)
            perror("Out of Space");
        else
        {
            T->Element = X;
            T->Left = T->Right = NULL;
        }
    }
    else if (X < T->Element)
        T->Left = Insert(X, T->Left);
    else
        T->Right = Insert(X, T->Right);

    return T;
}


SearchTree Delete(ElementType X, SearchTree T)
{
    Position TmpCell;

    if (T == NULL)
        perror("Element not found");
    // Found the element to be deleted
    else if (X < T->Left)
        T->Left = Delete(X, T->Left);
    else if(X > T-Right)
        T->Right = Delete(X, T->Right);

    // Two children
    else if (T->Left && T->Right)
    {
        TmpCell = FindMin(T->Right);
        T->Element = TmpCell->Element;
        T->Right = Delete(T->Element, T->Right);
    }
    // One or zero children
    else
    {
        TmpCell = T;
        if (T->Left == NULL)
            T = T->Right;
        else if (T->Right == NULL)
            T = T->Left;
        free(TmpCell);
    }

    return T;
}