【数据结构】中B树的插入,删除,遍历

桃扇骨 2022-06-16 10:14 167阅读 0赞

## 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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