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.youkuaiyun.com/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:weege@126.com
@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:weege@126.com
@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:weege@126.com
@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:weege@126.com
@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:weege@126.com
@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)
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