B-tree的代码实现 - c / c++ 版本

看到一篇相关的好文章，引用下：http://www.cnblogs.com/leoo2sk/archive/2011/07/10/mysql-index.html 。相当滴不错，备忘下。

在这篇文章中http://blog.csdn.net/weege/article/details/6526512介绍了B-tree/B+tree/B*tree,并且介绍了B-tree的查找，插入，删除操作。现在重新认识下B-TREE（温故而知新嘛~，确实如此。自己在写代码中会体会到，B-tree的操作出现的条件相对其他树比较复杂，调试也是一个理通思路的过程。)

B-tree又叫平衡多路查找树。一棵m阶的B-tree (m叉树)的特性如下：

（其中ceil(x)是一个取上限的函数）

1)  树中每个结点至多有m个孩子；

2)  除根结点和叶子结点外，其它每个结点至少有有ceil(m / 2)个孩子；

3)  若根结点不是叶子结点，则至少有2个孩子（特殊情况：没有孩子的根结点，即根结点为叶子结点，整棵树只有一个根节点）；

4)  所有叶子结点都出现在同一层，叶子结点不包含任何关键字信息(可以看做是外部结点或查询失败的结点，实际上这些结点不存在，指向这些结点的指针都为null)（PS:这种说法是按照严蔚敏那本教材给出的，具体操作不同而定，下面的实现中的叶子结点是树的终端结点，即没有孩子的结点）；

5)  每个非终端结点中包含有n个关键字信息： (n，P0，K1，P1，K2，P2，......，Kn，Pn)。其中：

             a)   Ki (i=1...n)为关键字，且关键字按顺序排序K(i-1)< Ki。

             b)   Pi为指向子树根的接点，且指针P(i-1)指向子树种所有结点的关键字均小于Ki，但都大于K(i-1)。

      c)   关键字的个数n必须满足： ceil(m / 2)-1 <= n <= m-1。

具体代码实现如下：(这里只是给出了简单的B-Tree结构，在内存中的数据操作。具体详情见代码吧~！）

头文件：（提供B-Tree基本的操作接口)

/***************************************************************************
	@coder:weedge E-mail:[email protected]
	@date:2011/08/27
	@comment:
		参考：http://www.cppblog.com/converse/archive/2009/10/13/98521.html
		实现对order序(阶)的B-TREE结构基本操作的封装。
		查找：search，插入：insert，删除：remove。
		创建：create，销毁：destory，打印：print。
**********************************************************/
#ifndef BTREE_H
#define BTREE_H

#ifdef __cplusplus
extern "C" {
#endif

////* 定义m序(阶)B 树的最小度数BTree_D=ceil(m/2)*/
/// 在这里定义每个节点中关键字的最大数目为:2 * BTree_D - 1，即序(阶)：2 * BTree_D.
#define BTree_D        2
#define ORDER        (BTree_D * 2) //定义为4阶B-tree,2-3-4树。(偶序)
//#define ORDER        (BTree_D * 2-1)//最简单为3阶B-tree,2-3树。(奇序)

	typedef int KeyType;
	typedef struct BTNode{
		int keynum;                        /// 结点中关键字的个数，ceil(ORDER/2)-1<= keynum <= ORDER-1
		KeyType key[ORDER-1];                /// 关键字向量为key[0..keynum - 1]
		struct BTNode* child[ORDER];        /// 孩子指针向量为child[0..keynum]
		char isLeaf;                    /// 是否是叶子节点的标志
	}BTNode;
	
	typedef BTNode* BTree;	///定义BTree
	
	///给定数据集data,创建BTree。
	void BTree_create(BTree* tree, const KeyType* data, int length);

	///销毁BTree，释放内存空间。
	void BTree_destroy(BTree* tree);
	
	///在BTree中插入关键字key。
	void BTree_insert(BTree* tree, KeyType key);

	///在BTree中移除关键字key。
	void BTree_remove(BTree* tree, KeyType key);

	///深度遍历BTree打印各层结点信息。
	void BTree_print(const BTree tree, int layer);
	
	/// 在BTree中查找关键字 key，
	/// 成功时返回找到的节点的地址及 key 在其中的位置 *pos
	/// 失败时返回 NULL 及查找失败时扫描到的节点位置 *pos
	BTNode* BTree_search(const BTree tree, int key, int* pos);
	
#ifdef __cplusplus
}
#endif

#endif

源文件：（提供B-Tree基本的基本操作的实现）

/***************************************************************************
	@coder:weedge E-mail:[email protected]
	@date:2011/08/27
	@comment:
		参考：http://www.cppblog.com/converse/archive/2009/10/13/98521.html
		实现对order序(阶)的B-TREE结构基本操作的封装。
		查找：search，插入：insert，删除：remove。
		创建：create，销毁：destory，打印：print。
**********************************************************/
#include <stdlib.h>
#include <stdio.h>
#include <assert.h>
#include "btree.h"

//#define max(a, b) (((a) > (b)) ? (a) : (b))
#define cmp(a, b) ( ( ((a)-(b)) >= (0) ) ? (1) : (0) ) //比较a，b大小
#define DEBUG_BTREE


// 模拟向磁盘写入节点
void disk_write(BTNode* node)
{
	int i;
//打印出结点中的全部元素，方便调试查看keynum之后的元素是否为0(即是否存在垃圾数据)；而不是keynum个元素。
    printf("向磁盘写入节点");
	for(i=0;i<ORDER-1;i++){
		printf("%c",node->key[i]);
	}
	printf("\n");
}

// 模拟从磁盘读取节点
void disk_read(BTNode** node)
{
	int i;
//打印出结点中的全部元素，方便调试查看keynum之后的元素是否为0(即是否存在垃圾数据)；而不是keynum个元素。
	printf("向磁盘读取节点");
	for(i=0;i<ORDER-1;i++){
		printf("%c",(*node)->key[i]);
	}
	printf("\n");
}

// 按层次打印 B 树
void BTree_print(const BTree tree, int layer)
{
    int i;
    BTNode* node = tree;

    if (node) {
        printf("第 %d 层， %d node : ", layer, node->keynum);

		//打印出结点中的全部元素，方便调试查看keynum之后的元素是否为0(即是否存在垃圾数据)；而不是keynum个元素。
        for (i = 0; i < ORDER-1; ++i) {
		//for (i = 0; i < node->keynum; ++i) {
            printf("%c ", node->key[i]);
        }

        printf("\n");

        ++layer;
        for (i = 0 ; i <= node->keynum; i++) {
            if (node->child[i]) {
                BTree_print(node->child[i], layer);
            }
        }
    }
    else {
        printf("树为空。\n");
    }
}

// 结点node内对关键字进行二分查找。
int binarySearch(BTNode* node, int low, int high, KeyType Fkey)
{
	int mid;
	while (low<=high)
	{
		mid = low + (high-low)/2;
		if (Fkey<node->key[mid])
		{
			high = mid-1;
		}
		if (Fkey>node->key[mid])
		{
			low = mid+1;
		}
		if (Fkey==node->key[mid])
		{
			return mid;//返回下标。
		}
	}
	return -1;//未找到返回-1.
}

//=======================================================insert=====================================

/***************************************************************************************
   将分裂的结点中的一半元素给新建的结点，并且将分裂结点中的中间关键字元素上移至父节点中。
   parent 是一个非满的父节点
   node 是 tree 孩子表中下标为 index 的孩子节点，且是满的，需分裂。
*******************************************************************/
void BTree_split_child(BTNode* parent, int index, BTNode* node)
{
	int i;
	BTNode* newNode;
#ifdef DEBUG_BTREE
	printf("BTree_split_child!\n");
#endif
    assert(parent && node);


    // 创建新节点，存储 node 中后半部分的数据
    newNode = (BTNode*)calloc(sizeof(BTNode), 1);
    if (!newNode) {
        printf("Error! out of memory!\n");
        return;
    }

    newNode->isLeaf = node->isLeaf;
    newNode->keynum = BTree_D - 1;

    // 拷贝 node 后半部分关键字,然后将node后半部分置为0。
    for (i = 0; i < newNode->keynum; ++i){
        newNode->key[i] = node->key[BTree_D + i];
        node->key[BTree_D + i] = 0;
    }

    // 如果 node 不是叶子节点，拷贝 node 后半部分的指向孩子节点的指针，然后将node后半部分指向孩子节点的指针置为NULL。
    if (!node->isLeaf) {
        for (i = 0; i < BTree_D; i++) {
            newNode->child[i] = node->child[BTree_D + i];
            node->child[BTree_D + i] = NULL;
        }
    }

    // 将 node 分裂出 newNode 之后，里面的数据减半
    node->keynum = BTree_D - 1;

    // 调整父节点中的指向孩子的指针和关键字元素。分裂时父节点增加指向孩子的指针和关键元素。
    for (i = parent->keynum; i > index; --i) {
        parent->child[i + 1] = parent->child[i];
    }

    parent->child[index + 1] = newNode;

    for (i = parent->keynum - 1; i >= index; --i) {
        parent->key[i + 1] = parent->key[i];
    }

    parent->key[index] = node->key[BTree_D - 1];
    ++parent->keynum;

	node->key[BTree_D - 1] = 0;

    // 写入磁盘
     disk_write(parent);
     disk_write(newNode);
     disk_write(node);
}

void BTree_insert_nonfull(BTNode* node, KeyType key)
{
	int i;
    assert(node);

    // 节点是叶子节点，直接插入
    if (node->isLeaf) {
        i = node->keynum - 1;
        while (i >= 0 && key < node->key[i]) {
            node->key[i + 1] = node->key[i];
            --i;
        }

        node->key[i + 1] = key;
        ++node->keynum;

        // 写入磁盘
        disk_write(node);
    }

    // 节点是内部节点
    else {
        /* 查找插入的位置*/
        i = node->keynum - 1;
        while (i >= 0 && key < node->key[i]) {
            --i;
        }

        ++i;

        // 从磁盘读取孩子节点
        disk_read(&node->child[i]);

        // 如果该孩子节点已满，分裂调整值
        if (node->child[i]->keynum == (ORDER-1)) {
            BTree_split_child(node, i, node->child[i]);
			// 如果待插入的关键字大于该分裂结点中上移到父节点的关键字，在该关键字的右孩子结点中进行插入操作。
            if (key > node->key[i]) {
                ++i;
            }
        }
        BTree_insert_nonfull(node->child[i], key);
    }
}

void BTree_insert(BTree* tree, KeyType key)
{
	BTNode* node;
    BTNode* root = *tree;

#ifdef DEBUG_BTREE
	printf("BTree_insert:\n");
#endif
    // 树为空
    if (NULL == root) {
        root = (BTNode*)calloc(sizeof(BTNode), 1);
        if (!root) {
            printf("Error! out of memory!\n");
            return;
        }
        root->isLeaf = 1;
        root->keynum = 1;
        root->key[0] = key;

        *tree = root;

        // 写入磁盘
        disk_write(root);

        return;
    }

    // 根节点已满，插入前需要进行分裂调整
    if (root->keynum == (ORDER-1)) {
        // 产生新节点当作根
        node = (BTNode*)calloc(sizeof(BTNode), 1);
        if (!node) {
            printf("Error! out of memory!\n");
            return;
        }

        *tree = node;
        node->isLeaf = 0;
        node->keynum = 0;
        node->child[0] = root;

        BTree_split_child(node, 0, root);

        BTree_insert_nonfull(node, key);
    }

    // 根节点未满，在当前节点中插入 key
    else {
        BTree_insert_nonfull(root, key);
    }
}

//=================================================remove========================================
/***********************************************************************************
// 对 tree 中的节点 node 进行合并孩子节点处理.
// 注意：孩子节点的 keynum 必须均已达到下限，即均等于 BTree_D - 1
// 将 tree 中索引为 index  的 key 下移至左孩子结点中，
// 将 node 中索引为 index + 1 的孩子节点合并到索引为 index 的孩子节点中，右孩子合并到左孩子结点中。
// 并调相关的 key 和指针。
***************************************************/
void BTree_merge_child(BTree* tree, BTNode* node, int index)
{
	int i;
	KeyType key;
	BTNode *leftChild, *rightChild;
#ifdef DEBUG_BTREE
	printf("BTree_merge_child!\n");
#endif
    assert(tree && node && index >= 0 && index < node->keynum);


    key = node->key[index];
    leftChild = node->child[index];
    rightChild = node->child[index + 1];

    assert(leftChild && leftChild->keynum == BTree_D - 1
        && rightChild && rightChild->keynum == BTree_D - 1);

	// 将 node中关键字下标为index 的 key 下移至左孩子结点中，该key所对应的右孩子结点指向node的右孩子结点中的第一个孩子。
    leftChild->key[leftChild->keynum] = key;
    leftChild->child[leftChild->keynum + 1] = rightChild->child[0];
    ++leftChild->keynum;

    // 右孩子的元素合并到左孩子结点中。
    for (i = 0; i < rightChild->keynum; ++i) {
        leftChild->key[leftChild->keynum] = rightChild->key[i];
        leftChild->child[leftChild->keynum + 1] = rightChild->child[i + 1];
        ++leftChild->keynum;
    }

    // 在 node 中下移的 key后面的元素前移
    for (i = index; i < node->keynum - 1; ++i) {
        node->key[i] = node->key[i + 1];
        node->child[i + 1] = node->child[i + 2];
    }
    node->key[node->keynum - 1] = 0;
    node->child[node->keynum] = NULL;
    --node->keynum;

    // 如果根节点没有 key 了，并将根节点调整为合并后的左孩子节点；然后删除释放空间。
    if (node->keynum == 0) {
        if (*tree == node) {
            *tree = leftChild;
        }

        free(node);
        node = NULL;
    }

    free(rightChild);
	rightChild = NULL;
}

void BTree_recursive_remove(BTree* tree, KeyType key)
{
    // B-数的保持条件之一：
    // 非根节点的内部节点的关键字数目不能少于 BTree_D - 1

    int i, j, index;
    BTNode *root = *tree;
    BTNode *node = root;

    if (!root) {
        printf("Failed to remove %c, it is not in the tree!\n", key);
        return;
    }

	// 结点中找key。
    index = 0;
    while (index < node->keynum && key > node->key[index]) {
        ++index;
    }

/*======================含有key的当前结点时的情况====================
node:
index of Key:			 i-1  i  i+1
						+---+---+---+---+
						  *  key   *
					+---+---+---+---+---+
						   /     \
index of Child:		      i	     i+1
					     /		   \
					+---+---+	   +---+---+
				      *   *			  *   *   
				+---+---+---+  +---+---+---+
				    leftChild	  rightChild
============================================================*/
    /*一、结点中找到了关键字key的情况.*/
    if (index < node->keynum && node->key[index] == key) {
		BTNode *leftChild, *rightChild;
		KeyType leftKey, rightKey;
        /* 1，所在节点是叶子节点，直接删除*/
        if (node->isLeaf) {
            for (i = index; i < node->keynum-1; ++i) {
                node->key[i] = node->key[i + 1];
                //node->child[i + 1] = node->child[i + 2];叶子节点的孩子结点为空，无需移动处理。
            }
			node->key[node->keynum-1] = 0;
			//node->child[node->keynum] = NULL;
            --node->keynum;

            if (node->keynum == 0) {
                assert(node == *tree);
                free(node);
                *tree = NULL;
            }

            return;
        }
		/*2.选择脱贫致富的孩子结点。*/
		// 2a，选择相对富有的左孩子结点。
        // 如果位于 key 前的左孩子结点的 key 数目 >= BTree_D，
        // 在其中找 key 的左孩子结点的最后一个元素上移至父节点key的位置。
        // 然后在左孩子节点中递归删除元素leftKey。
        else if (node->child[index]->keynum >= BTree_D) {
            leftChild = node->child[index];
            leftKey = leftChild->key[leftChild->keynum - 1];
            node->key[index] = leftKey;

            BTree_recursive_remove(&leftChild, leftKey);
        }
		// 2b，选择相对富有的右孩子结点。
        // 如果位于 key 后的右孩子结点的 key 数目 >= BTree_D，
        // 在其中找 key 的右孩子结点的第一个元素上移至父节点key的位置
        // 然后在右孩子节点中递归删除元素rightKey。
        else if (node->child[index + 1]->keynum >= BTree_D) {
            rightChild = node->child[index + 1];
            rightKey = rightChild->key[0];
            node->key[index] = rightKey;

            BTree_recursive_remove(&rightChild, rightKey);
        }
		/*左右孩子结点都刚脱贫。删除前需要孩子结点的合并操作*/
        // 2c，左右孩子结点只包含 BTree_D - 1 个节点，
        // 合并是将 key 下移至左孩子节点，并将右孩子节点合并到左孩子节点中，
        // 删除右孩子节点，在父节点node中移除 key 和指向右孩子节点的指针，
        // 然后在合并了的左孩子节点中递归删除元素key。
        else if (node->child[index]->keynum == BTree_D - 1
            && node->child[index + 1]->keynum == BTree_D - 1){
            leftChild = node->child[index];

            BTree_merge_child(tree, node, index);

            // 在合并了的左孩子节点中递归删除 key
            BTree_recursive_remove(&leftChild, key);
        }
    }

/*======================未含有key的当前结点时的情况====================
node:
index of Key:			 i-1  i  i+1
						+---+---+---+---+
						  *  keyi *
					+---+---+---+---+---+
					   /    |    \
index of Child:		 i-1	i     i+1
					 /		|       \
		    +---+---+	+---+---+	   +---+---+
		     *   *	      *   *		     *   *   
	    +---+---+---+   +---+---+---+  +---+---+---+
		leftSibling		  Child	       rightSibling	
============================================================*/
	/*二、结点中未找到了关键字key的情况.*/
    else {
		BTNode *leftSibling, *rightSibling, *child;
		// 3. key 不在内节点 node 中，则应当在某个包含 key 的子节点中。
		//  key < node->key[index], 所以 key 应当在孩子节点 node->child[index] 中
        child = node->child[index];
        if (!child) {
            printf("Failed to remove %c, it is not in the tree!\n", key);
            return;
        }
		/*所需查找的该孩子结点刚脱贫的情况*/
        if (child->keynum == BTree_D - 1) {
            leftSibling = NULL;
            rightSibling = NULL;

            if (index - 1 >= 0) {
                leftSibling = node->child[index - 1];
            }

            if (index + 1 <= node->keynum) {
                rightSibling = node->child[index + 1];
            }
			/*选择致富的相邻兄弟结点。*/
            // 3a，如果所在孩子节点相邻的兄弟节点中有节点至少包含 BTree_D 个关键字
            // 将 node 的一个关键字key[index]下移到 child 中，将相对富有的相邻兄弟节点中一个关键字上移到
            // node 中，然后在 child 孩子节点中递归删除 key。
            if ((leftSibling && leftSibling->keynum >= BTree_D)
                || (rightSibling && rightSibling->keynum >= BTree_D)) {
				int richR = 0;
				if(rightSibling) richR = 1;
				if(leftSibling && rightSibling) {
					richR = cmp(rightSibling->keynum,leftSibling->keynum);
				}
                if (rightSibling && rightSibling->keynum >= BTree_D && richR) {
					//相邻右兄弟相对富有，则该孩子先向父节点借一个元素，右兄弟中的第一个元素上移至父节点所借位置，并进行相应调整。
                    child->key[child->keynum] = node->key[index];
                    child->child[child->keynum + 1] = rightSibling->child[0];
                    ++child->keynum;

                    node->key[index] = rightSibling->key[0];

                    for (j = 0; j < rightSibling->keynum - 1; ++j) {//元素前移
                        rightSibling->key[j] = rightSibling->key[j + 1];
                        rightSibling->child[j] = rightSibling->child[j + 1];
                    }
					rightSibling->key[rightSibling->keynum-1] = 0;
					rightSibling->child[rightSibling->keynum-1] = rightSibling->child[rightSibling->keynum];
					rightSibling->child[rightSibling->keynum] = NULL;
                    --rightSibling->keynum;
                }
                else {//相邻左兄弟相对富有，则该孩子向父节点借一个元素，左兄弟中的最后元素上移至父节点所借位置，并进行相应调整。
                    for (j = child->keynum; j > 0; --j) {//元素后移
                        child->key[j] = child->key[j - 1];
                        child->child[j + 1] = child->child[j];
                    }
                    child->child[1] = child->child[0];
                    child->child[0] = leftSibling->child[leftSibling->keynum];
                    child->key[0] = node->key[index - 1];
                    ++child->keynum;

                    node->key[index - 1] = leftSibling->key[leftSibling->keynum - 1];

					leftSibling->key[leftSibling->keynum - 1] = 0;
					leftSibling->child[leftSibling->keynum] = NULL;

                    --leftSibling->keynum;
                }
            }
			/*相邻兄弟结点都刚脱贫。删除前需要兄弟结点的合并操作,*/
            // 3b, 如果所在孩子节点相邻的兄弟节点都只包含 BTree_D - 1 个关键字，
            // 将 child 与其一相邻节点合并，并将 node 中的一个关键字下降到合并节点中，
            // 再在 node 中删除那个关键字和相关指针，若 node 的 key 为空，删之，并调整根为合并结点。
            // 最后，在相关孩子节点child中递归删除 key。
            else if ((!leftSibling || (leftSibling && leftSibling->keynum == BTree_D - 1))
                && (!rightSibling || (rightSibling && rightSibling->keynum == BTree_D - 1))) {
                if (leftSibling && leftSibling->keynum == BTree_D - 1) {

                    BTree_merge_child(tree, node, index - 1);//node中的右孩子元素合并到左孩子中。

                    child = leftSibling;
                }

                else if (rightSibling && rightSibling->keynum == BTree_D - 1) {

                    BTree_merge_child(tree, node, index);//node中的右孩子元素合并到左孩子中。
                }
            }
        }

        BTree_recursive_remove(&child, key);//调整后，在key所在孩子结点中继续递归删除key。
    }
}

void BTree_remove(BTree* tree, KeyType key)
{
#ifdef DEBUG_BTREE
	printf("BTree_remove:\n");
#endif
	if (*tree==NULL)
	{	
		printf("BTree is NULL!\n");
		return;
	}

	BTree_recursive_remove(tree, key);
}

//=====================================search====================================

BTNode* BTree_recursive_search(const BTree tree, KeyType key, int* pos)
{
    int i = 0;

    while (i < tree->keynum && key > tree->key[i]) {
        ++i;
    }

    // Find the key.
    if (i < tree->keynum && tree->key[i] == key) {
        *pos = i;
        return tree;
    }

    // tree 为叶子节点，找不到 key，查找失败返回
    if (tree->isLeaf) {
        return NULL;
    }

    // 节点内查找失败，但 tree->key[i - 1]< key < tree->key[i]，
    // 下一个查找的结点应为 child[i]

    // 从磁盘读取第 i 个孩子的数据
    disk_read(&tree->child[i]);

    // 递归地继续查找于树 tree->child[i]
    return BTree_recursive_search(tree->child[i], key, pos);
}

BTNode* BTree_search(const BTree tree, KeyType key, int* pos)
{
#ifdef DEBUG_BTREE
	printf("BTree_search:\n");
#endif
	if (!tree) {
		printf("BTree is NULL!\n");
        return NULL;
    }
	*pos = -1;
	return BTree_recursive_search(tree,key,pos);
}

//===============================create===============================
void BTree_create(BTree* tree, const KeyType* data, int length)
{
	int i, pos = -1;
    assert(tree);

#ifdef DEBUG_BTREE
    printf("\n 开始创建 B-树，关键字为:\n");
    for (i = 0; i < length; i++) {
        printf(" %c ", data[i]);
    }
    printf("\n");
#endif

    for (i = 0; i < length; i++) {
#ifdef DEBUG_BTREE
        printf("\n插入关键字 %c:\n", data[i]);
#endif

		BTree_search(*tree,data[i],&pos);//树的递归搜索。
		if (pos!=-1)
		{
			printf("this key %c is in the B-tree,not to insert.\n",data[i]);
		}else{
			BTree_insert(tree, data[i]);//插入元素到BTree中。
		}

#ifdef DEBUG_BTREE
        BTree_print(*tree,1);//树的深度遍历,从第一层开始。
#endif
    }

    printf("\n");
}
//===============================destroy===============================
void BTree_destroy(BTree* tree)
{
    int i;
    BTNode* node = *tree;

    if (node) {
        for (i = 0; i <= node->keynum; i++) {
            BTree_destroy(&node->child[i]);
        }

        free(node);
    }

    *tree = NULL;
}

测试文件：（测试B-Tree基本的操作接口)

/***************************************************************************
	@coder:weedge E-mail:[email protected]
	@date:2011/08/28
	@comment:
		测试order序(阶)的B-TREE结构基本操作。
		查找：search，插入：insert，删除：remove。
		创建：create，销毁：destory，打印：print。
**********************************************************/

#include <stdio.h>
#include "btree.h"

void test_BTree_search(BTree tree, KeyType key)
{
    int pos = -1;
    BTNode*    node = BTree_search(tree, key, &pos);
    if (node) {
        printf("在%s节点（包含 %d 个关键字）中找到关键字 %c，其索引为 %d\n",
            node->isLeaf ? "叶子" : "非叶子",
            node->keynum, key, pos);
    }
    else {
        printf("在树中找不到关键字 %c\n", key);
    }
}

void test_BTree_remove(BTree* tree, KeyType key)
{
    printf("\n移除关键字 %c \n", key);
    BTree_remove(tree, key);
    BTree_print(*tree);
    printf("\n");
}

void test_btree()
{

    KeyType array[] = {
        'G','G', 'M', 'P', 'X', 'A', 'C', 'D', 'E', 'J', 'K',
        'N', 'O', 'R', 'S', 'T', 'U', 'V', 'Y', 'Z', 'F', 'X'
    };
    const int length = sizeof(array)/sizeof(KeyType);
    BTree tree = NULL;
    BTNode* node = NULL;
    int pos = -1;
    KeyType key1 = 'R';        // in the tree.
    KeyType key2 = 'B';        // not in the tree.

    // 创建
    BTree_create(&tree, array, length);

    printf("\n=== 创建 B- 树 ===\n");
    BTree_print(tree);
    printf("\n");

    // 查找
    test_BTree_search(tree, key1);
    printf("\n");
    test_BTree_search(tree, key2);

	// 移除不在B树中的元素
	test_BTree_remove(&tree, key2);
	printf("\n");

    // 插入关键字
    printf("\n插入关键字 %c \n", key2);
    BTree_insert(&tree, key2);
    BTree_print(tree);
    printf("\n");

    test_BTree_search(tree, key2);

    // 移除关键字
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

    key2 = 'M';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

    key2 = 'E';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

    key2 = 'G';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

    key2 = 'A';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

    key2 = 'D';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

    key2 = 'K';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

    key2 = 'P';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

    key2 = 'J';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

    key2 = 'C';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

    key2 = 'X';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

	key2 = 'O';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

	key2 = 'V';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

	key2 = 'R';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

	key2 = 'U';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

	key2 = 'T';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

	key2 = 'N';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);
	key2 = 'S';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);
	key2 = 'Y';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);
	key2 = 'F';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);
	key2 = 'Z';
    test_BTree_remove(&tree, key2);
    test_BTree_search(tree, key2);

	// 销毁
	BTree_destroy(&tree);
}

int main()
{
	test_btree();

	return 0;
}

另外参考《Data.structures.and.Program.Design.in.Cpp》-Section 11.3:EXTERNALSEARCHING:B-TREES的讲解实现，这边书个人认为比较经典，如果对数据结构和算法比较感兴趣的话，可以作为参考读物，不错的，根据数据结构上的操作与程序上的实现相结合，讲的很细。这本书好像没有中文版的，即使有，也推荐看原版吧，毕竟写代码都是用英文字符，实现也比较贴切易懂。去网上找这本书的资料还挺多的。而且国内有些大学也参考这本书讲解数据结构和算法。比如：http://sist.sysu.edu.cn/~isslxm/DSA/CS09/。

头文件：（采用C++模板（template）来实现B-Tree基本的操作接口)

/***************************************************************************
	@editer:weedge E-mail:[email protected]
	@date:2011/08/27
	@comment:
		Data.structures.and.Program.Design.in.Cpp
		Section 11.3:EXTERNALSEARCHING:B-TREES
		采用泛型编程（模板template），Record为关键字类型，order为序(阶)，
		实现对order序(阶)的B-TREE结构基本操作的封装。
		查找：search，插入：insert，删除：remove。
**********************************************************/

#ifndef B_Tree_H_
#define B_Tree_H_

enum Error_code{overflow=-2,duplicate_error=-1,not_present=0,success=1};

template <class Record, int order>
struct B_node {
///  data members:
   int count;
   Record data[order - 1];
   B_node<Record, order> *branch[order];///在大多数应用中，这些指针被不同的磁盘中块(block)的地址代替。
///  constructor:
   B_node();
};

template <class Record, int order>
class B_tree {
public:  
	////*  Add public methods. */

	Error_code search_tree(Record &target);

	Error_code insert(const Record &new_entry);

	Error_code remove(const Record &target);

protected:
	////*  data members */
	
	////*  Add protected auxiliary functions here in order to inherit for subclass. */
	///=============================search=========================================

	Error_code recursive_search_tree(B_node<Record, order> *current, Record &target);
	
	Error_code search_node(B_node<Record, order> *current, const Record &target, int &position);

	///=============================insert=========================================

	Error_code push_down(B_node<Record, order> *current,const Record &new_entry,
		Record &median,B_node<Record, order> *&right_branch);
	
	void push_in(B_node<Record, order> *current, const Record &entry, 
		B_node<Record, order> *right_branch, int position);
	
	void split_node(B_node<Record, order> *current,	const Record &extra_entry, B_node<Record, order> *extra_branch,
		int position, B_node<Record, order> *&right_half, Record &median);

	///==============================remove========================================

	Error_code recursive_remove(B_node<Record, order> *current, const Record &target);
	
	void remove_data(B_node<Record, order> *current, int position);

	void copy_in_predecessor(B_node<Record, order> *current, int position);
	
	void restore(B_node<Record, order> *current,int position);
	
	void move_left(B_node<Record, order> *current, int position);
	
	void move_right(B_node<Record, order> *current,int position);
	
	void combine(B_node<Record, order> *current, int position);

private: 
	////*  data members */
	B_node<Record, order> *root;

    ////*  Add private auxiliary functions here. */
};

#endif //end B_Tree_H_

原文件：（实现B-Tree基本的基本操作)

/***************************************************************************
	@editor:weedge E-mail:[email protected]
	@date:2011/08/27
	@comment:
		Data.structures.and.Program.Design.in.Cpp
		Section 11.3:EXTERNALSEARCHING:B-TREES
		采用泛型编程（模板template），
		实现对B-TREE结构基本操作的封装。
		查找：search，插入：insert，删除：remove。
**********************************************************/
#include "B_Tree.h"

template <class Record, int order>
Error_code B_tree<Record, order>::search_tree(Record &target)
/*
Post: If there is an entry in the B-tree whose key matches that in target,
      the parameter target is replaced by the corresponding Record from
      the B-tree and a code of success is returned.  Otherwise
      a code of not_present is returned.
Uses: recursive_search_tree
*/
{
   return recursive_search_tree(root, target);
}


template <class Record, int order>
Error_code B_tree<Record, order>::recursive_search_tree(
           B_node<Record, order> *current, Record &target)
/*
Pre:  current is either NULL or points to a subtree of the B_tree.
Post: If the Key of target is not in the subtree, a code of not_present
      is returned. Otherwise, a code of success is returned and
      target is set to the corresponding Record of the subtree.
Uses: recursive_search_tree recursively and search_node
*/
{
   Error_code result = not_present;
   int position;
   if (current != NULL) {
      result = search_node(current, target, position);
      if (result == not_present)
         result = recursive_search_tree(current->branch[position], target);
      else
         target = current->data[position];
   }
   return result;
}


template <class Record, int order>
Error_code B_tree<Record, order>::search_node(
   B_node<Record, order> *current, const Record &target, int &position)
/*
Pre:  current points to a node of a B_tree.
Post: If the Key of target is found in *current, then a code of
      success is returned, the parameter position is set to the index
      of target, and the corresponding Record is copied to
      target.  Otherwise, a code of not_present is returned, and
      position is set to the branch index on which to continue the search.
Uses: Methods of class Record.
*/
{
   position = 0;
   while (position < current->count && target > current->data[position])
      position++;         //  Perform a sequential search through the keys.
   if (position < current->count && target == current->data[position])
      return success;
   else
      return not_present;
}


template <class Record, int order>
Error_code B_tree<Record, order>::insert(const Record &new_entry)
/*
Post: If the Key of new_entry is already in the B_tree,
      a code of duplicate_error is returned.
      Otherwise, a code of success is returned and the Record new_entry
      is inserted into the B-tree in such a way that the properties of a B-tree
      are preserved.
Uses: Methods of struct B_node and the auxiliary function push_down.
*/
{
   Record median;
   B_node<Record, order> *right_branch, *new_root;
   Error_code result = push_down(root, new_entry, median, right_branch);

   if (result == overflow) {  //  The whole tree grows in height.
                              //  Make a brand new root for the whole B-tree.
      new_root = new B_node<Record, order>;
      new_root->count = 1;
      new_root->data[0] = median;
      new_root->branch[0] = root;
      new_root->branch[1] = right_branch;
      root = new_root;
      result = success;
   }
   return result;
}


template <class Record, int order>
Error_code B_tree<Record, order>::push_down(
                 B_node<Record, order> *current,
                 const Record &new_entry,
                 Record &median,
                 B_node<Record, order> *&right_branch)
/*
Pre:  current is either NULL or points to a node of a B_tree.
Post: If an entry with a Key matching that of new_entry is in the subtree
      to which current points, a code of duplicate_error is returned.
      Otherwise, new_entry is inserted into the subtree: If this causes the
      height of the subtree to grow, a code of overflow is returned, and the
      Record median is extracted to be reinserted higher in the B-tree,
      together with the subtree right_branch on its right.
      If the height does not grow, a code of success is returned.
Uses: Functions push_down (called recursively), search_node,
      split_node, and push_in.
*/
{
   Error_code result;
   int position;
   if (current == NULL) { //  Since we cannot insert in an empty tree, the recursion terminates.
      median = new_entry;
      right_branch = NULL;
      result = overflow;
   }
   else {   //   Search the current node.
      if (search_node(current, new_entry, position) == success)
         result = duplicate_error;
      else {
         Record extra_entry;
         B_node<Record, order> *extra_branch;
         result = push_down(current->branch[position], new_entry,
                            extra_entry, extra_branch);
         if (result == overflow) {  //  Record extra_entry now must be added to current
            if (current->count < order - 1) {
               result = success;
               push_in(current, extra_entry, extra_branch, position);
            }

            else split_node(current, extra_entry, extra_branch, position,
                            right_branch, median);
            //  Record median and its right_branch will go up to a higher node.
         }
      }
   }
   return result;
}


template <class Record, int order>
void B_tree<Record, order>::push_in(B_node<Record, order> *current,
   const Record &entry, B_node<Record, order> *right_branch, int position)
/*
Pre:  current points to a node of a B_tree.  The node *current is not full
      and entry belongs in *current at index position.
Post: entry has been inserted along with its right-hand branch
      right_branch into *current at index position.
*/
{
   for (int i = current->count; i > position; i--) {  //  Shift all later data to the right.
      current->data[i] = current->data[i - 1];
      current->branch[i + 1] = current->branch[i];
   }
   current->data[position] = entry;
   current->branch[position + 1] = right_branch;
   current->count++;
}


template <class Record, int order>
void B_tree<Record, order>::split_node(
   B_node<Record, order> *current,    //  node to be split
   const Record &extra_entry,          //  new entry to insert
   B_node<Record, order> *extra_branch,//  subtree on right of extra_entry
   int position,                  //  index in node where extra_entry goes
   B_node<Record, order> *&right_half, //  new node for right half of entries
   Record &median)                     //  median entry (in neither half)
/*
Pre:  current points to a node of a B_tree.
      The node *current is full, but if there were room, the record
      extra_entry with its right-hand pointer extra_branch would belong
      in *current at position position, 0 <= position < order.
Post: The node *current with extra_entry and pointer extra_branch at
      position position are divided into nodes *current and *right_half
      separated by a Record median.
Uses: Methods of struct B_node, function push_in.
*/
{
   right_half = new B_node<Record, order>;
   int mid = order/2;  //  The entries from mid on will go to right_half.
   if (position <= mid) {   //  First case:  extra_entry belongs in left half.
      for (int i = mid; i < order - 1; i++) {  //  Move entries to right_half.
         right_half->data[i - mid] = current->data[i];
         right_half->branch[i + 1 - mid] = current->branch[i + 1];
      }
      current->count = mid;
      right_half->count = order - 1 - mid;
      push_in(current, extra_entry, extra_branch, position);
   }
   else {  //  Second case:  extra_entry belongs in right half.
      mid++;      //  Temporarily leave the median in left half.
      for (int i = mid; i < order - 1; i++) {  //  Move entries to right_half.
         right_half->data[i - mid] = current->data[i];
         right_half->branch[i + 1 - mid] = current->branch[i + 1];
      }
      current->count = mid;
      right_half->count = order - 1 - mid;
      push_in(right_half, extra_entry, extra_branch, position - mid);
   }
      median = current->data[current->count - 1]; //  Remove median from left half.
      right_half->branch[0] = current->branch[current->count];
      current->count--;
}


template <class Record, int order>
Error_code B_tree<Record, order>::remove(const Record &target)
/*
Post: If a Record with Key matching that of target belongs to the
      B_tree, a code of success is returned and the corresponding node
      is removed from the B-tree.  Otherwise, a code of not_present
      is returned.
Uses: Function recursive_remove
*/
{
   Error_code result;
   result = recursive_remove(root, target);
   if (root != NULL && root->count == 0) {  //  root is now empty.
      B_node<Record, order> *old_root = root;
      root = root->branch[0];
      delete old_root;
   }
   return result;
}


template <class Record, int order>
Error_code B_tree<Record, order>::recursive_remove(
   B_node<Record, order> *current, const Record &target)
/*
Pre:  current is either NULL or
      points to the root node of a subtree of a B_tree.
Post: If a Record with Key matching that of target belongs to the subtree,
      a code of success is returned and the corresponding node is removed
      from the subtree so that the properties of a B-tree are maintained.
      Otherwise, a code of not_present is returned.
Uses: Functions search_node, copy_in_predecessor,
      recursive_remove (recursively), remove_data, and restore.
*/
{
   Error_code result;
   int position;
   if (current == NULL) result = not_present;
   else {
      if (search_node(current, target, position) == success) {  //  The target is in the current node.
         result = success;
         if (current->branch[position] != NULL) {     //  not at a leaf node
            copy_in_predecessor(current, position);

            recursive_remove(current->branch[position],
                             current->data[position]);
         }
         else remove_data(current, position);     //  Remove from a leaf node.
      }
      else result = recursive_remove(current->branch[position], target);
      if (current->branch[position] != NULL)
         if (current->branch[position]->count < (order - 1) / 2)
            restore(current, position);
   }
   return result;
}


template <class Record, int order>
void B_tree<Record, order>::remove_data(B_node<Record, order> *current,
                                        int position)
/*
Pre:  current points to a leaf node in a B-tree with an entry at position.
Post: This entry is removed from *current.
*/
{
   for (int i = position; i < current->count - 1; i++)
      current->data[i] = current->data[i + 1];
   current->count--;
}


template <class Record, int order>
void B_tree<Record, order>::copy_in_predecessor(
                    B_node<Record, order> *current, int position)
/*
Pre:  current points to a non-leaf node in a B-tree with an entry at position.
Post: This entry is replaced by its immediate predecessor under order of keys.
*/
{
   B_node<Record, order> *leaf = current->branch[position];  //   First go left from the current entry.
   while (leaf->branch[leaf->count] != NULL)
      leaf = leaf->branch[leaf->count]; //   Move as far rightward as possible.
   current->data[position] = leaf->data[leaf->count - 1];
}


template <class Record, int order>
void B_tree<Record, order>::restore(B_node<Record, order> *current,
                                    int position)
/*
Pre:  current points to a non-leaf node in a B-tree; the node to which
      current->branch[position] points has one too few entries.
Post: An entry is taken from elsewhere to restore the minimum number of
      entries in the node to which current->branch[position] points.
Uses: move_left, move_right, combine.
*/
{
   if (position == current->count)   //  case:  rightmost branch
      if (current->branch[position - 1]->count > (order - 1) / 2)
         move_right(current, position - 1);
      else
         combine(current, position);
   else if (position == 0)       //  case: leftmost branch
      if (current->branch[1]->count > (order - 1) / 2)
         move_left(current, 1);
      else
         combine(current, 1);
   else                          //  remaining cases: intermediate branches
      if (current->branch[position - 1]->count > (order - 1) / 2)
         move_right(current, position - 1);
      else if (current->branch[position + 1]->count > (order - 1) / 2)
         move_left(current, position + 1);
      else
         combine(current, position);
}


template <class Record, int order>
void B_tree<Record, order>::move_left(B_node<Record, order> *current,
                                      int position)
/*
Pre:  current points to a node in a B-tree with more than the minimum
      number of entries in branch position and one too few entries in branch
      position - 1.
Post: The leftmost entry from branch position has moved into
      current, which has sent an entry into the branch position - 1.
*/
{
   B_node<Record, order> *left_branch = current->branch[position - 1],
                         *right_branch = current->branch[position];
   left_branch->data[left_branch->count] = current->data[position - 1];  //  Take entry from the parent.
   left_branch->branch[++left_branch->count] = right_branch->branch[0];
   current->data[position - 1] = right_branch->data[0];  //   Add the right-hand entry to the parent.
   right_branch->count--;
   for (int i = 0; i < right_branch->count; i++) {   //  Move right-hand entries to fill the hole.
      right_branch->data[i] = right_branch->data[i + 1];
      right_branch->branch[i] = right_branch->branch[i + 1];
   }
   right_branch->branch[right_branch->count] =
      right_branch->branch[right_branch->count + 1];
}


template <class Record, int order>
void B_tree<Record, order>::move_right(B_node<Record, order> *current,
                                       int position)
/*
Pre:  current points to a node in a B-tree with more than the minimum
      number of entries in branch position and one too few entries
      in branch position + 1.
Post: The rightmost entry from branch position has moved into
      current, which has sent an entry into the branch position + 1.
*/
{
   B_node<Record, order> *right_branch = current->branch[position + 1],
                         *left_branch = current->branch[position];
   right_branch->branch[right_branch->count + 1] =
      right_branch->branch[right_branch->count];
   for (int i = right_branch->count ; i > 0; i--) {  //  Make room for new entry.
      right_branch->data[i] = right_branch->data[i - 1];
      right_branch->branch[i] = right_branch->branch[i - 1];
   }
   right_branch->count++;
   right_branch->data[0] = current->data[position]; //  Take entry from parent.
   right_branch->branch[0] = left_branch->branch[left_branch->count--];
   current->data[position] = left_branch->data[left_branch->count];
}


template <class Record, int order>
void B_tree<Record, order>::combine(B_node<Record, order> *current,
                                    int position)
/*
Pre:  current points to a node in a B-tree with entries in the branches
      position and position - 1, with too few to move entries.
Post: The nodes at branches position - 1 and position have been combined
      into one node, which also includes the entry formerly in current at
      index  position - 1.
*/
{
   int i;
   B_node<Record, order> *left_branch = current->branch[position - 1],
                         *right_branch = current->branch[position];
   left_branch->data[left_branch->count] = current->data[position - 1];
   left_branch->branch[++left_branch->count] = right_branch->branch[0];
   for (i = 0; i < right_branch->count; i++) {
      left_branch->data[left_branch->count] = right_branch->data[i];
      left_branch->branch[++left_branch->count] =
                                        right_branch->branch[i + 1];
   }
   current->count--;
   for (i = position - 1; i < current->count; i++) {
      current->data[i] = current->data[i + 1];
      current->branch[i + 1] = current->branch[i + 2];
   }
   delete right_branch;
}

测试文件：（待写)