二叉查找樹，也稱作二叉搜索樹，有序二叉樹，排序二叉樹，而當(dāng)一棵空樹或者具有下列性質(zhì)的二叉樹，就可以被定義為二叉查找樹：

• 若任意節(jié)點(diǎn)的左子樹不空，則左子樹上所有節(jié)點(diǎn)的值均小于它的根節(jié)點(diǎn)的值。

• 若任意節(jié)點(diǎn)的右子樹不空，則右子樹上所有節(jié)點(diǎn)的值均大于它的根節(jié)點(diǎn)的值。

• 任意節(jié)點(diǎn)的左、右子樹也分別為二叉查找樹。

• 沒有鍵值相等的節(jié)點(diǎn)。

二叉查找樹相比于其他數(shù)據(jù)結(jié)構(gòu)的優(yōu)勢在查找、插入的時間復(fù)雜度較低，為O(log n)。二叉查找樹是基礎(chǔ)性數(shù)據(jù)結(jié)構(gòu)，用于構(gòu)建更為抽象的數(shù)據(jù)結(jié)構(gòu),如集合、multiset、關(guān)聯(lián)數(shù)組等。對于大量的輸入數(shù)據(jù)，鏈表的線性訪問時間太慢，不宜使用。

下面來看我們?yōu)槎娌檎覙涠x的抽象行為：

#ifndef _Tree_H

struct TreeNode;
typedef struct TreeNode *Position;
typedef struct TreeNode *SearchTree;
typedef int ElementType;

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

#endif

而對于上述抽象行為的實現(xiàn)，我們先來給出實現(xiàn)代碼:

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "Tree.h"

#define OK 1
#define ERROR 0
#define TRUE 1
#define FALSE 0

typedef int Status;

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)
    {

        /* Create and return a one-node tree */
        T = malloc(sizeof( struct TreeNode ));
        if ( T == NULL )
            printf("Out of space!!!\n");
        else
        {
            T->Element = X;
            T->Left = T->Right = NULL;
        }
    }
    else if (X < T->Element)
        T->Left = Insert(X, T->Left);
    else if (X > T->Element)
        T->Right = Insert(X, T->Right);
    /* Else X is in the tree already; we'll do nothing */

    return T;
}

SearchTree Delete(ElementType X, SearchTree T)
{
    Position TmpCell;
    if (T == NULL)
        printf("Element not found\n");
    else if (X < T->Element) /* Go left */
        T->Right = Delete(X, T->Left);
    else if (X > T->Element) /* Go Right */
        T->Right = Delete(X, T->Left);
    else if (T->Left && T->Right) /* Two Children */
    {
        /* Replace with smallest in right subtree */
        TmpCell = FindMin(T->Right);
        T->Element = TmpCell->Element;
        T->Right = Delete(T->Element, T->Right);
    }
    else /* One or zero children */
    {
        TmpCell = T;
        if (T->Left == NULL) /* Also handles 0 children */
            T = T->Right;
        else if (T->Right == NULL)
            T = T->Left;
        free( TmpCell );
    }

    return T;
}

ElementType Retrieve(Position P)
{
    return P->Element;
}

/**
 * 前序遍歷"二叉樹"
 * @param T Tree
 */
void PreorderTravel(SearchTree T)
{
    if (T != NULL)
    {
        printf("%d\n", T->Element);
        PreorderTravel(T->Left);
        PreorderTravel(T->Right);
    }
}

/**
 * 中序遍歷"二叉樹"
 * @param T Tree
 */
void InorderTravel(SearchTree T)
{
    if (T != NULL)
    {
        InorderTravel(T->Left);
        printf("%d\n", T->Element);
        InorderTravel(T->Right);
    }
}

/**
 * 后序遍歷二叉樹
 * @param T Tree
 */
void PostorderTravel(SearchTree T)
{
    if (T != NULL)
    {
        PostorderTravel(T->Left);
        PostorderTravel(T->Right);
        printf("%d\n", T->Element);
    }
}

void PrintTree(SearchTree T, ElementType Element, int direction)
{
    if (T != NULL)
    {
        if (direction == 0)
            printf("%2d is root\n", T->Element);
        else
            printf("%2d is %2d's %6s child\n", T->Element, Element, direction == 1 ? "right" : "left");

        PrintTree(T->Left, T->Element, -1);
        PrintTree(T->Right, T->Element, 1);
    }
}

最后我們對我們的實現(xiàn)代碼，在main函數(shù)中進(jìn)行測試:

int main(int argc, char const *argv[])
{
    printf("Hello Leon\n");
    SearchTree T;
    MakeEmpty(T);

    T = Insert(21, T);
    T = Insert(2150, T);
    T = Insert(127, T);
    T = Insert(121, T);

    printf("樹的詳細(xì)信息: \n");
    PrintTree(T, T->Element, 0);

    printf("前序遍歷二叉樹: \n");
    PreorderTravel(T);

    printf("中序遍歷二叉樹: \n");
    InorderTravel(T);

    printf("后序遍歷二叉樹: \n");
    PostorderTravel(T);

    printf("最大值: %d\n", FindMax(T)->Element);
    printf("最小值: %d\n", FindMin(T)->Element);

    return 0;
}

編譯運(yùn)行這個C文件，控制臺打印的信息如下:

Hello wsx
樹的詳細(xì)信息:
21 is root
2150 is 21's  right child
127 is 2150's   left child
121 is 127's   left child
前序遍歷二叉樹:
21
2150
127
121
中序遍歷二叉樹:
21
121
127
2150
后序遍歷二叉樹:
121
127
2150
21
最大值: 2150
最小值: 21

測試成功。