【算法导论】二叉搜索树

二叉搜索树的基本操作与树的高度成正比，而我们并不能保证随机地构造二叉树，所以又有了较稳定的二叉树的变体，比如红黑树、B树、B+树、平衡树等。二叉搜索树则是理解其他变体的基础，所以想对它基本操作进行一次系统的学习了。

二叉搜索树的概念：其左子结点小于或等于该节点，其右结点大于或等于该节点。

祖先的概念：

二叉搜索树树的遍历：

中序遍历：先输出左子树关键字，然后输出该结点关键字，最后输出右结点关键字。

先序遍历：先输出该结点关键字，然后输出左子树关键字，最后输出右结点关键字。

后续遍历：先输出左结点关键字，然后输出右节点关键字，最后输出该结点关键字。

中序遍历实现代码

template<typename T>
class binarySearchNode
{
public:
    binarySearchNode(T value,binarySearchNode<T> * leftBinarySearchNode=null,binarySearchNode<T>* rightBinarySearchNode=null, binarySearchNode<T>* father=null){
        this.value=value;
        this.leftBinarySearchNode=leftBinarySearchNode;
        this.rightBinarySearchNode=rightBinarySearchNode;
        this.father=father;
    }
    ~binarySearchNode();

    /* data */
private:
    T value;
    binarySearchNode<T>* leftBinarySearchNode;
    binarySearchNode<T>* rightBinarySearchNode;
    binarySearchNode<T>* father;    
};

template<typename T>
class binarySearchTree
{
public:
    binarySearchTree(binarySearchNode<T>* head){
        this.head=head;
    };
    ~binarySearchTree();

    /* data */

public:
    std::vector<T> inorderTreeWalk(binarySearchNode* x);
    binarySearchNode* treeSearch(binarySearchNode* k);
    binarySearchNode* treeMinmum();
    binarySearchNode* treeMaxmum();
    binarySearchNode* treeSuccessor(binarySearchNode* x);
    binarySearchNode* treePredecessor(binarySearchNode* x);
    void treeInsert(binarySearchNode* x);
    void treeDelete(binarySearchNode* x);
    friend void transplant(binarySearchNode* u, binarySearchNode* v);
private:
    binarySearchNode<T>* head;

};

T* binarySearchTree::inorderTreeWalk(){

}

binarySearchNode* binarySearchTree::treeMinmum(){
    binarySearchNode* cur=head;
    if(cur->leftBinarySearchNode!=null){
        cur=cur->leftBinarySearchNode;
    }
    return cur;
}
binarySearchNode* binarySearchTree::treeMaxmum(){
    binarySearchNode* cur=head;
    if(cur->rightBinarySearchNode!=null){
        cur=cur->rightBinarySearchNode;
    }
    return cur;
}
binarySearchNode* binarySearchTree::treeSuccessor(binarySearchNode* x){

    if(x->rightBinarySearchNode!=null){
        return treeMinmum(x->rightBinarySearchNode);
    }
    binarySearchNode* y=x->father;

    while(y!=null&&y==x->rightBinarySearchNode){
        x=y;
        y=x->father;
    }
    return y;
}
binarySearchNode* binarySearchTree::treePredecessor(){
    if(x->leftBinarySearchNode!=null){
        return treeMaxmum(x->leftBinarySearchNode);
    }
    binarySearchNode* y=x->father;
    while(y!=null&&x==y.leftBinarySearchNode){
        x=y;
        y=x->father;
    }
    return y;
}
std::vector<T> binarySearchTree::inorderTreeWalk(binarySearchNode* x){
    std::vector<T> result;

    if(x!=null){
        inorderTreeWalk(x->leftBinarySearchNode);
        result.push_back(x->value);
        inorderTreeWalk(x->rightBinarySearchNode);
    }
}
binarySearchNode* binarySearchTree::treeSearch(binarySearchNode* cur,T k){
    if(k==cur->data){
        return cur;
    }else if(k<cur->data{
        return treeSearch(cur->leftBinarySearchNode,T k);
    }else{
        return treeSearch(cur->rightBinarySearchNode,T k);
    }
}

void binarySearchTree::treeInsert(binarySearchNode* k){
    binarySearchNode* cur=head;

    while(cur!=null){
        if(k->data<cur->data){
            cur=cur->leftBinarySearchNode;
        }else{
            cur=cur->rightBinarySearchNode;
        }
    }

    k->father=cur;

    if(cur==null){
        this->head=cur;
    }else if(k->data>cur->value){
        cur->rightBinarySearchNode=k;
    }else{
        cur->leftBinarySearchNode=k;
    }
}

void binarySearchTree::treeDelete(binarySearchNode* x){
    if(x->leftBinarySearchNode==null)
        transplant(x,x->rightBinarySearchNode);
    elseif(x->rightBinarySearchNode==null){
        transplant(x,x->leftBinarySearchNode);
    }else{
        binarySearchNode* rChild=x->rightBinarySearchNode;
        if(rChild->leftBinarySearchNode==null){
            transplant(x,rChild);
            rChild->leftBinarySearchNode=x->leftBinarySearchNode;
            x->leftBinarySearchNode->father=rChild;
        }else{
            binarySearchNode* cur=treeMinmum(x->leftBinarySearchNode);\
            if(cur->leftBinarySearchNode!=null)
                cur->leftBinarySearchNode->father=cur->father;
            transplant(x,cur);
            cur->leftBinarySearchNode=x->leftBinarySearchNode;
            x->leftBinarySearchNode->father=cur;
        }
    }
}
friend void transplant(binarySearchNode* u, binarySearchNode* v){
    if(u->father==null)
        v->father=head;
    else if(u=u->father->leftBinarySearchNode)
        u->father->leftBinarySearchNode=v;
    else
        u->father->rightBinarySearchNode=v;

    if(v!=null)
        v->father=u->father;
}

12.2-6 如果二叉搜索树T的一个结点x的右子树为空，且x有一个后继y，那么y一定是x最底层的祖先，并且其左孩子也是x的祖先。