B树的插入、删除与遍历

B树的插入

定义：
1、根节点至少有两个分支
2、除了根节点以外，所有节点的关键字个数至少为M/2个，最多为M-1
3、每个节点的度数均是关键字数加一
4、所有的叶子节点都在同一层
插入：
我们设计节点的结构如下：

#define M 5#define  MAX M - 1#define MIN M/2typedef char KeyType;typedef struct {}Record;typedef struct ElemType{    KeyType key;    Record *recptr;}ElemType;typedef struct BNode{    int num;    BNode *parent;    ElemType data[M+1];    BNode*sub[M+1];}BNode,*BTree;//查找函数的返回值类型typedef struct Result{    bool tag;    BNode*pnode;    int index;}Result;

插入代码：
当以个节点的个数大于MAX时就分裂，如果根节点分裂会产生新根，否则就将分裂出来的节点插入到双亲中，如果双亲又大于MAX就继续分裂，这样就能保证B树的定义的正确性，代码如下：

ElemType MoveItem(BNode*ptr, BNode *s, int pos){    int tmp = ptr->num;    for (int i = pos + 1, j = 0; i <= tmp; i++, j++)    {        s->data[j] = ptr->data[i];        s->sub[j] = ptr->sub[i];        if (ptr->sub[i] != NULL)        {            s->sub[j]->parent = s;        }    }    s->parent = ptr->parent;    s->num = ptr->num = MIN;    return s->data[0];}BNode * MakeRoot(ElemType x, BNode *left, BNode *right){    BNode *s = BuyNode();    s->num = 1;    s->data[1] = x;    s->sub[0] = left;    if (left != NULL)        left->parent = s;    s->sub[1] = right;    if (right != NULL)        right->parent = s;    return s;}bool InsertItem(BNode*ptr, int pos, ElemType e, BNode*right);BNode *Splice(BNode*ptr){    BNode *s = BuyNode();    ElemType e = MoveItem(ptr, s, MIN);    if (ptr->parent == NULL)    {        return MakeRoot(e,ptr, s);    }    else    {        ptr = ptr->parent;        int i = ptr->num;        ptr->data[0] = e;//这句很关键，如果ptr->data[0]未设置就会和0位置比较还没有结果，插入位置就会出错        while (ptr->data[i].key > s->data[0].key) --i;        InsertItem(ptr, i + 1, s->data[0], s);        if (ptr->num > MAX)        {            return Splice(ptr);        }        return NULL;    }}bool InsertItem(BNode*ptr, int pos, ElemType e, BNode*right)//BNode&node{    for (int i = ptr->num; i >= pos; --i)    {        ptr->data[i + 1] = ptr->data[i];        ptr->sub[i + 1] = ptr->sub[i];    }    //    ptr->data[pos] = e;    ptr->sub[pos] = right;    if (right != NULL)    {        right->parent = ptr;    }//    ptr->num += 1;    return true;}bool Insert(BTree *ptr, ElemType e){    if (ptr == NULL)        return false;    if (*ptr == NULL)    {        *ptr = MakeRoot(e, NULL, NULL);        return true;    }    Result res=FindValue(*ptr, e.key);    if (res.pnode == NULL || res.tag) return false;    InsertItem(res.pnode, res.index+1, e, NULL);    if (res.pnode->num > MAX)    {        BNode*p = Splice(res.pnode);        if (p != NULL)        {            *ptr = p;        }    }    return true;}

辅助函数：

BNode* BuyNode(){    BNode *node = new BNode();    if (node == NULL)        exit(-1);    memset(node, 0, sizeof(BNode));    return node;}Result FindValue(BNode*ptr, KeyType e){    Result res = { false, NULL, -1 };    while (ptr != NULL)    {        int i = ptr->num;        ptr->data[0].key = e;        while (ptr->data[i].key > e) --i;        res.pnode = ptr;        res.index = i;        if (i != 0 && ptr->data[i].key == e)        {            res.tag = true;            break;        }        else            ptr = ptr->sub[i];    }    return res;}

B树的删除

B树的删除，我们将带有分支的节点中的关键码删除，用他的前驱和后继替换掉这个被删除的关键码，然后删除前驱或者后继，删除前驱或者后继之后，会出现与B树定义不相符的情况，比如关键码个数小于MIN的情况，这个时候就要做相应的旋转，如过旋转不了就只有进行节点的合并，合并有可能会产生新根，代码如下：

//找前驱BNode *FindPre(BNode*ptr){    while (ptr!=NULL&&ptr->sub[ptr->num] != NULL)    {        ptr = ptr->sub[ptr->num];    }    return ptr;}//找后继BNode *FindNext(BNode*ptr){    while (ptr != NULL&&ptr->sub[0] != NULL)    {        ptr = ptr->sub[0];    }    return ptr;}//删除叶子结点void DelLeafItem(BNode *ptr, int pos){    for (int i = pos; i < ptr->num; i++)    {        ptr->data[i] = ptr->data[i + 1];        ptr->sub[i] = ptr->sub[i + 1];    }    ptr->num -= 1;}//右旋转void RightRotateLeaf(BNode *leftbro, BNode*ptr, BNode *parent, int pos){    ptr->data[0] = parent->data[pos];    for (int i = ptr->num; i >= 0; i--)    {        ptr->data[i + 1] = ptr->data[i];        ptr->sub[i + 1] = ptr->sub[i];    }    ptr->num += 1;    ptr->sub[0] = leftbro->sub[leftbro->num];    if (ptr->sub[0] != NULL)//    {        ptr->sub[0]->parent = ptr;    }    parent->data[pos] = leftbro->data[leftbro->num];    leftbro->num -= 1;}//左旋转void LeftRotateLeaf(BNode *rightbro,BNode *ptr,BNode *parent,int pos){    ptr ->data[ptr->num+1] = parent->data[pos + 1];    ptr->sub[ptr->num + 1] = rightbro->sub[0];    if (ptr->sub[ptr->num+1]!=NULL)    {        ptr->sub[ptr->num + 1]->parent = ptr;    }    ptr->num += 1;    parent->data[pos + 1] = rightbro->data[1];    for (int i =0; i < rightbro->num; i++)    {        rightbro->data[i] = rightbro->data[i + 1];        rightbro->sub[i] = rightbro->sub[i + 1];    }    rightbro->num -= 1;}//向左合并void LeftMerge(BNode*leftbro, BNode*ptr, BNode*parent, int pos){    ptr->data[0] = parent->data[pos];    for (int i = 0,j=leftbro->num+1; i <= ptr->num; i++,j++)    {        leftbro->data[j] = ptr->data[i];        leftbro->sub[j] = ptr->sub[i];        if (leftbro->sub[j] != NULL)        {            leftbro->sub[j]->parent = leftbro;        }    }    leftbro->num = leftbro->num + ptr->num + 1;    free(ptr);    DelLeafItem(parent, pos);}//向右合并void RightMerge(BNode *ptr, BNode *rightbro, BNode *parent, int pos){     LeftMerge(ptr, rightbro, parent, pos+1);}//出现小于MIN的情况的调整函数BNode *AdjusLeaf(BNode*ptr){    BNode*parent = ptr->parent;    int pos = 0;    while (parent->sub[pos] != ptr) ++pos;    BNode*leftbro = pos-1<0?NULL:parent->sub[pos-1];    BNode*rightbro = pos+1>=MAX?NULL:parent->sub[pos+1];    if (leftbro!=NULL&&leftbro->num>MIN)    {        RightRotateLeaf(leftbro,ptr,parent,pos);    }    else if (rightbro!=NULL&&rightbro->num>MIN)    {        LeftRotateLeaf(rightbro, ptr,parent, pos);    }    else if(leftbro!=NULL)    {        LeftMerge(leftbro, ptr, parent, pos);        ptr = leftbro;    }    else if (rightbro != NULL)    {         RightMerge(ptr, rightbro, parent, pos);        // ptr = rightbro;    }    if (parent->parent != NULL&&parent->num < MIN)    {        return AdjusLeaf(parent);    }    if (parent->parent == NULL&&parent->num <= 0)    {        free(parent);        ptr->parent = NULL;        return ptr;    }    return NULL;}//删除函数void ReMove(BNode*&root, KeyType e){    if (root == NULL)        return;    Result res = FindValue(root, e);    if (res.pnode == NULL || res.tag==false) return;    BNode *ptr = res.pnode;    int pos = res.index;    BNode*pre = FindPre(ptr->sub[pos-1]);    BNode*next = FindNext(ptr->sub[pos]);    if (pre != NULL&&pre->num > MIN)    {        ptr->data[pos] = pre->data[pre->num];        ptr = pre;        pos = pre->num;    }    else if (next != NULL&&next->num > MIN)    {        ptr->data[pos] = next->data[1];        ptr = next;        pos = 1;    }    else if (pre != NULL)    {        ptr->data[pos] = pre->data[pre->num];        ptr = pre;        pos = pre->num;    }    else if (next != NULL)    {        ptr->data[pos] = next->data[1];        ptr = next;        pos = 1;    }    DelLeafItem(ptr, pos);//    if (ptr->parent != NULL&&ptr->num < MIN)    {        BNode*newroot = AdjusLeaf(ptr);        if (newroot != NULL)        {            root = newroot;        }    }    else if (ptr->parent == NULL&&ptr->num <= 0)    {        free(root);        root = NULL;    }}

B树的遍历

利用递归的特性 ，代码十分简洁，先递归到最左边，然后打印一个关键码就遍历一个分支，代码如下：

void InOder(BNode*root){    if (root != NULL)    {        InOder(root->sub[0]);        for (int i = 1; i <= root->num; i++)        {            cout << root->data[i].key;            InOder(root->sub[i]);        }    }}