红黑树的概念以及性质
概念：
红黑树是一颗二叉搜索树
每个节点不是红色，就是黑色
最长路径最多是最短路径的二倍
2 红黑树的性质
1. 每个结点不是红色就是黑色
2. 根节点是黑色的
3. 如果一个节点是红色的，则它的两个孩子结点是黑色的
4. 对于每个结点，从该结点到其所有后代叶结点的简单路径上，均 包含相同数目的黑色结点
5. 每个叶子结点都是黑色的(此处的叶子结点指的是空结点)
为什么满足上面的性质，红黑树就能保证：其最长路径中节点个数不会超过最短路径节点个数的两倍？
红黑树节点的定义
enum Colour { RED, BLACK }; template<class K> struct rbtreenode { rbtreenode<K>* _left; rbtreenode<K>* _right; rbtreenode<K>* _parent; K _key; Colour _col; rbtreenode(const K& key) :_left(nullptr) , _right(nullptr) , _parent(nullptr)//节点的双亲(红黑树需要旋转，为了实现简单给 出该字段) , _key(key) , _col(RED) // 节点的颜色 {} };
思考：在节点的定义中，为什么要将节点的默认颜色给成红色的？
这是因为方便新插入的节点的颜色
那我们新插入的结点应该选择插入红色还是黑色呢？？
对于这个问题，如果插入的黑色节点，会打破哪个规则，如果插入的是红色节点又会打破哪个规则?
如果插入黑色节点，需要打破规则4，要检查每个路径很难
如果插入红色节点，需要打破规则3，这时候只需要处理父亲是否为红
所以插入最好是红色，所以初始化列表的颜色为红色
红黑树插入
红黑树也是二叉搜索树，插入可以复用之前的代码
bool insert(const K& key) { if (_root == nullptr) { _root = new node(key); _root->_col = BLACK; return true; } node* cur = _root; node* parent = nullptr; while (cur) { if (cur->_key > key) { parent = cur; cur = cur->_left; } else if (cur->_key < key) { parent = cur; cur = cur->_right; } else { return false; } } cur = new node(key); if (parent->_key > key) { parent->_left = cur; } else { parent->_right = cur; } cur->_parent = parent; //红黑树插入处理颜色逻辑 }
红黑树插入处理颜色逻辑
红黑树插入操作详解（一）
如果父亲节点为黑色的话就不用管了，如果插入的结点父亲是红色，此时违背了规则3，此时需要进行处理
约定:cur为当前节点，p为父节点，g为祖父节点，u为叔叔节点
情况一: cur为红，p为红，g为黑，u存在且为红 (g为根节点)
解决方式：将p,u改为黑，g改为红
情况二: cur为红，p为红，g为黑，u存在且为红 (g不为根节点)
解决方式：将p,u改为黑，g改为红，然后把g当成cur，继续向上调整。
红黑树插入操作详解（二）
情况三: cur为红，p为红，g为黑，u不存在/u存在且为黑
p为g的左孩子，cur为p的左孩子，则进行右单旋转；p、g变色–p变黑，g变红
红黑树插入操作详解（三）
情况四: cur为红，p为红，g为黑，u不存在/u存在且为黑
p为g的左孩子，cur为p的右孩子，则针对p做左单旋转，则转换成了情况3
代码实现插入
bool insert(const K& key) { if (_root == nullptr) { _root = new node(key); _root->_col = BLACK; return true; } node* cur = _root; node* parent = nullptr; while (cur) { if (cur->_key > key) { parent = cur; cur = cur->_left; } else if (cur->_key < key) { parent = cur; cur = cur->_right; } else { return false; } } cur = new node(key); if (parent->_key > key) { parent->_left = cur; } else { parent->_right = cur; } cur->_parent = parent; while (parent && parent->_col == RED) { node* grandfather = parent->_parent; if (parent == grandfather->_left) { node* uncle = grandfather->_right; if (uncle && uncle->_col == RED) { parent->_col = BLACK; uncle->_col = BLACK; grandfather->_col = RED; cur = grandfather; parent = cur->_parent; } else { if (cur == parent->_right) { spinright(grandfather); parent->_col = BLACK; grandfather->_col = RED; } else { spinleft(parent); spinright(grandfather); cur->_col = BLACK;//这里注意情况3转情况4的时候cur会对应情况3的p,情况3的p会变成黑色，这里我们将cur变成黑色即可 grandfather->_col = RED; } break; } } } _root->_col = BLACK;//为了保险起见，最后直接把根节点变黑 return true; }
针对以上情况，可能出现左右镜像的情况，逻辑一模一样，把上面所以关于方向的变量用他的相反代替，就得到
bool insert(const K& key) { if (_root == nullptr) { _root = new node(key); _root->_col = BLACK; return true; } node* cur = _root; node* parent = nullptr; while (cur) { if (cur->_key > key) { parent = cur; cur = cur->_left; } else if (cur->_key < key) { parent = cur; cur = cur->_right; } else { return false; } } cur = new node(key); if (parent->_key > key) { parent->_left = cur; } else { parent->_right = cur; } cur->_parent = parent; while (parent && parent->_col == RED) { node* grandfather = parent->_parent; if (parent == grandfather->_left) { node* uncle = grandfather->_right; if (uncle && uncle->_col == RED) { parent->_col = BLACK; uncle->_col = BLACK; grandfather->_col = RED; cur = grandfather; parent = cur->_parent; } else { if (cur == parent->_right) { spinright(grandfather); parent->_col = BLACK; grandfather->_col = RED; } else { spinleft(parent); spinright(grandfather); cur->_col = BLACK; grandfather->_col = RED; } break; } } else { node* uncle = grandfather->_left; if (uncle && uncle->_col == RED) { parent->_col = BLACK; uncle->_col = BLACK; grandfather->_col = RED; cur = grandfather; parent = cur->_parent; } else { if (cur == parent->_left) { spinleft(grandfather); parent->_col = BLACK; grandfather->_col = RED; } else { spinright(parent); spinleft(grandfather); cur->_col = BLACK; grandfather->_col = RED; } break; } } } _root->_col = BLACK; return true; }
红黑树的验证
我们需要验证根如果是红色就不是红黑树，并且如果相邻的两个节点是红色的话，也不是红黑树
bool Check(node* cur) { if (cur == nullptr) return true; if (cur->_col == RED && cur->_parent->_col == RED)//如果要判断当前节点和子节点的话，一个节点要判断两次，所以和父亲结点比较，只需一次 { cout << cur->_key<< "存在连续的红色节点" << endl; return false; } return Check(cur->_left) && Check(cur->_right); } bool IsBalance() { if (_root && _root->_col == RED) return false; return Check(_root); }
整体代码：
#include<iostream> #include<string> using namespace std; enum Colour { RED, BLACK }; template<class K> struct rbtreenode { rbtreenode<K>* _left; rbtreenode<K>* _right; rbtreenode<K>* _parent; K _key; Colour _col; rbtreenode(const K& key) :_left(nullptr) , _right(nullptr) , _parent(nullptr) , _key(key) , _col(RED) {} }; template<class K> class rbtree { typedef rbtreenode<K> node; public: void spinleft(node* parent) { node* subr = parent->_right; node* subrl = subr->_left; parent->_right = subrl; if (subrl) subrl->_parent = parent; subr->_left = parent; node* ppnode = parent->_parent; parent->_parent = subr; if (ppnode == nullptr) { _root = subr; subr->_parent = nullptr; } else { if (ppnode->_left == parent) { ppnode->_left = subr; subr->_parent = ppnode; } else { ppnode->_right = subr; subr->_parent = ppnode; } } } void spinright(node* parent) { node* subl = parent->_left; node* sublr = subl->_right; parent->_left = sublr; if (sublr) sublr->_parent = parent; subl->_right = parent; node* ppnode = parent->_parent; parent->_parent = subl; if (ppnode == nullptr) { _root = subl; subl->_parent = nullptr; } else { if (ppnode->_left == parent) { ppnode->_left = subl; subl->_parent = ppnode; } else { ppnode->_right = subl; subl->_parent = ppnode; } } } void spinlr(node* parent) { node* subl = parent->_left; node* sublr = subl->_right; //int bf = sublr->_bf; spinleft(parent->_left); spinright(parent); /* if (bf == 1) { subl->_bf = -1; sublr->_bf = 0; parent->_bf = 0; } else if (bf == -1) { subl->_bf = 0; sublr->_bf = 0; parent->_bf = 1; } else { subl->_bf = 0; sublr->_bf = 0; parent->_bf = 0; }*/ } void spinrl(node* parent) { node* subr = parent->_right; node* subrl = subr->_left; //int bf = subrl->_bf; spinright(subr); spinleft(parent); /* if (bf == 1) { parent->_bf = -1; subr->_bf = 0; subrl->_bf = 0; } else if (bf == -1) { parent->_bf = 0; subr->_bf = 1; subrl->_bf = 0; } else { subr->_bf = 0; subrl->_bf = 0; parent->_bf = 0; }*/ } bool insert(const K& key) { if (_root == nullptr) { _root = new node(key); _root->_col = BLACK; return true; } node* cur = _root; node* parent = nullptr; while (cur) { if (cur->_key > key) { parent = cur; cur = cur->_left; } else if (cur->_key < key) { parent = cur; cur = cur->_right; } else { return false; } } cur = new node(key); if (parent->_key > key) { parent->_left = cur; } else { parent->_right = cur; } cur->_parent = parent; while (parent && parent->_col == RED) { node* grandfather = parent->_parent; if (parent == grandfather->_left) { node* uncle = grandfather->_right; if (uncle && uncle->_col == RED) { parent->_col = BLACK; uncle->_col = BLACK; grandfather->_col = RED; cur = grandfather; parent = cur->_parent; } else { if (cur == parent->_right) { spinright(grandfather); parent->_col = BLACK; grandfather->_col = RED; } else { spinleft(parent); spinright(grandfather); cur->_col = BLACK; grandfather->_col = RED; } break; } } else { node* uncle = grandfather->_left; if (uncle && uncle->_col == RED) { parent->_col = BLACK; uncle->_col = BLACK; grandfather->_col = RED; cur = grandfather; parent = cur->_parent; } else { if (cur == parent->_left) { spinleft(grandfather); parent->_col = BLACK; grandfather->_col = RED; } else { spinright(parent); spinleft(grandfather); cur->_col = BLACK; grandfather->_col = RED; } break; } } } _root->_col = BLACK; return true; } bool Check(node* cur) { if (cur == nullptr) return true; if (cur->_col == RED && cur->_parent->_col == RED) { cout << cur->_key<< "存在连续的红色节点" << endl; return false; } return Check(cur->_left) && Check(cur->_right); } bool IsBalance() { if (_root && _root->_col == RED) return false; return Check(_root); } void inorder() { _inorder(_root); } void _inorder(node* root) { if (root == nullptr) return; _inorder(root->_left); cout << root->_key << endl; _inorder(root->_right); } node* Find(const int& key) { node* cur = _root; node* parent = nullptr; while (cur) { if (cur->_key > key) { parent = cur; cur = cur->_left; } else if (cur->_key < key) { parent = cur; cur = cur->_right; } else { return cur; } } return nullptr; } private: int treeheight(node* root) { if (root == nullptr) return 0; int leftheight = treeheight(root->_left); int rightheight = treeheight(root->_right); return leftheight > rightheight ? leftheight + 1 : rightheight + 1; } node* _root = nullptr; }; int main() { rbtree<int>st; int a[] = //{ 16,3,1 };//测试右旋 //{ 16,32,58 };//测试左旋 //{ 8,3,1,10,6,4};//测试右左旋 { 4, 2, 6, 1, 3, 5, 15, 7, 16,14 }; for (auto e : a) { st.insert(e); } st.inorder(); int ret = st.IsBalance(); cout << endl; cout << ret; }
