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/2
typedef 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]);
        }
    }
}