【数据结构】中B树的插入,删除,遍历
B树的定义：
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;
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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;
}
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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;
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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;
}
}
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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]);
}
}
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