该文章详细介绍了如何在C++中实现AVL树的插入功能，包括AVLTreeNode节点定义、AVLTree类结构、四种旋转操作（左单旋、右单旋、左右双旋、右左双旋）以及插入函数的具体实现。同时，文章还提供了测试AVLTree是否平衡的函数，确保插入后树的平衡性。
C++之AVL树的插入实现
文章目录
1.引言
众所周知，在我们使用普通的搜索二叉树的时候，难免会出现插入的数据总是过大或者过小，从而导致歪脖子树的情况出现，由此情况下AVL树便运营而生。AVLTree是一种在普通搜索二叉树上升级改造之后趋近于绝对平衡的结构，构建AVLTree的一种方法是引入平衡因子的方法，使得每一个左右子树的高度差的绝对值都不超过2，也就是只会有1，0，-1的这三种情况。
一般情况下，一棵AVLTree树应该具有：
1.它的左右子树都是AVL树
2.左右子树高度之差（平衡因子）绝对值不超过1
2.具体实现
整个过程我们会实现：
1.AVLTreeNode节点定义
2.AVLTree类定义
3.四种旋转的实现
4.AVLTree类中Insert()函数定义
5.测试AVLTree是否成功函数
2.1AVLTree节点定义
整个节点我们采用三叉链结构（左、右、父亲指针）和使用KV模型，并且最重要的是每个节点都会有一个bf平衡因子的整数衡量。
template<class K,class V>
struct AVLTreeNode
{
AVLTreeNode<K, V>* _left;//左指针
AVLTreeNode<K, V>* _right;//右指针
AVLTreeNode<K, V>* _parent;//双亲指针
pair<K, V> _kv;//存放数据的kv的pair对
int _bf;//平衡因子
AVLTreeNode(const pair<K,V>& kv) //构造函数
:_left(nullptr)
,_right(nullptr)
,_parent(nullptr)
,_kv(kv)
,_bf(0)
{}
};
2.1 AVLTree的类结构定义
整个类：
public：我们需要一个Insert()插入函数、走中序Inorder函数、判高度Height()函数和判断平衡Isbalance()函数
private：_root根节点与四种旋转函数
template<class K,class V>
class AVLTree
{
typedef AVLTreeNode<K,V> Node;
public:
bool Insert(const pair<K, V>& kv) {}
void Inorder() {}
int _Height(){}
bool Isbalance(){}
private:
bool _Isbalance(Node* root)
{}
int _Height(Node* root)
{}
void _Inorder(Node* root)
{}
void RotateL(Node* parent)
{}
void RotateR(Node* parent)
{}
void RotateLR(Node* parent)
{}
void RotateRL(Node* parent)
{}
2.2四种旋转实现
首先我们要知道我们为什么要旋转？ 之前我们提到了，当平衡因子不超过2的时候，才能称之为AVLTree，不过在插入的过程中，总会有一次使得平衡因子绝对值等于2，那么这个时候我们就需要旋转来降低树的高度了。而旋转具体又能分为四种旋转：
1.左单旋 2.右单旋 3.左右双旋 4.右左双旋
2.2.1新节点插入较高右子树的右侧–左单旋
void RotateL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL = subR->_left;
parent->_right = subRL;
if (subRL)
subRL->_parent = parent;
Node* pparent = parent->_parent;
subR->_left = parent;
parent->_parent = subR;
//判断双亲的双亲节点pparent
if (pparent == nullptr)
{
_root = subR;
_root->_parent = nullptr;
}
else
{
if (pparent->_left == parent)
{
pparent->_left = subR;
}
else
{
pparent->_right = subR;
}
subR->_parent = pparent;
}
parent->_bf = subR->_bf = 0;
}
2.2.2新节点插入较高左子树的右侧–右单旋
void RotateR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR = subL->_right;
parent->_left = subLR;
if (subLR)
subLR->_parent = parent;
Node* pparent = parent->_parent;
subL->_right = parent;
parent->_parent = subL;
if (pparent == nullptr)
{
_root = subLR;
_root->_parent = nullptr;
}
else
{
if (pparent->_left == parent)
{
pparent->_left = subL;
}
else
{
pparent->_right = subL;
}
subL->_parent = pparent;
}
parent->_bf = subL->_bf = 0;
}
2.2.3新节点插入较高左子树的右侧–左右双旋
void RotateLR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR = subL->_right;
int bf = subLR->_bf;
RotateL(parent->_left);
RotateR(parent);
if (bf == 0)
{
subLR->_bf = 0;
parent->_bf = 0;
subL->_bf = 0;
}
else if (bf == -1)
{
parent->_bf = 1;
subLR->_bf = 0;
subL->_bf = 0;
}
else if (bf == 1)
{
parent->_bf = 0;
subL->_bf = -1;
subLR->_bf = 0;
}
else
{
assert(false);
}
}
2.2.4新节点插入较高右子树的左侧–右左双旋
、
void RotateRL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL = parent->_left;
int bf = subRL->_bf;
RotateR(parent->_right);
RotateL(parent);
if (bf == 0)
{
parent->_bf = 0;
subR->_bf = 0;
subRL->_bf = 0;
}
else if (bf == -1)
{
parent->_bf = 0;
subR->_bf = 1;
subRL->_bf = 0;
}
else if (bf == 1)
{
subR->_bf = 0;
parent->_bf = -1;
subRL->_bf = 0;
}
else
{
assert(false);
}
}
2.3 Insert插入函数的具体实现
关键思路：在插入后当前最近插入节点的平衡因子为0时，不进行处理
在插入后当前最近插入节点的平衡因子为1或者-1时，需要处理往上更新；
在插入后当前最近插入节点的平衡因子为2或者-2时，需要旋转处理
bool Insert(const pair<K, V>& kv)
{
if (_root == nullptr) //为空root即为新增节点
{
_root = new Node(kv);
return true;
}
Node* parent = nullptr;
Node* cur = _root;
while (cur) //走到最下边
{
if (cur->_kv.first > kv.first)
{
parent = cur;
cur = cur->_left;
}
else if (cur->_kv.first < kv.first)
{
parent = cur;
cur = cur->_right;
}
else
{
return false;
}
}
cur = new Node(kv); //创建新节点链接
if (parent->_kv.first > kv.first)
{
parent->_left = cur;
}
else
{
parent->_right = cur;
}
cur->_parent = parent;
//更新平衡因子
while (parent)
{
if (cur == parent->_left)
{
parent->_bf--;
}
else
{
parent->_bf++;
}
//判断
if (parent->_bf == 1 || parent->_bf == -1)
{
parent = parent->_parent;
cur = cur->_parent;
}
else if (parent->_bf == 2 || parent->_bf == -2)
{
if (parent->_bf == 2 && cur->_bf == 1)
{
RotateL(parent);//左单旋
}
else if (parent->_bf == -2 && cur->_bf == -1)
{
RotateR(parent);//右单旋
}
else if (parent->_bf == -2 && cur->_bf == 1)
{
RotateLR(parent);//左右双旋
}
else if (parent->_bf == 2 && cur->_bf == -1)
{
RotateRL(parent);//右左双旋
}
else
{
assert("false");
}
break;
}
else if (parent->_bf == 0)//bf==0，不做处理
{
break;
}
else
{
assert("false");
}
}
return true;
}
2.4测试AVLTree是否成功
测试的思路是先通过Height高度函数算出AVLTree的左右子树高度，随后递归检测左右子树的高度差，若有一个不满足即驳回。
int _Height(Node* root) //高度函数
{
if (root == NULL)
return 0;
int leftH = _Height(root->_left);
int rightH = _Height(root->_right);
return leftH > rightH ? leftH + 1 : rightH + 1;
}
bool _Isbalance(Node* root)//判断平衡函数
{
if (root == NULL)
return true;
int leftH = _Height(root->_left);
int rightH = _Height(root->_right);
if (rightH - leftH != root->_bf)
{
cout << root->_kv.first << ":" << "平衡因子异常" << endl;
return false;
}
return abs(leftH - rightH) < 2
&& _Isbalance(root->_left)
&& _Isbalance(root->_right);
}
2.6 测试结果
测试代码
void testAVLTree()
{
AVLTree<int, int> lt;
lt.Insert(make_pair(1,1));
lt.Insert(make_pair(2,2));
lt.Insert(make_pair(7,7));
lt.Insert(make_pair(3,3));
lt.Insert(make_pair(4,4));
lt.Inorder();
cout << endl;
cout<<lt._Height();
cout << endl;
cout<<lt.Isbalance();
}
2.7 全部代码
#pragma once
namespace Arthur
{
template<class K,class V>
struct AVLTreeNode
{
AVLTreeNode<K, V>* _left;
AVLTreeNode<K, V>* _right;
AVLTreeNode<K, V>* _parent;
pair<K, V> _kv;
int _bf;
AVLTreeNode(const pair<K,V>& kv)
:_left(nullptr)
,_right(nullptr)
,_parent(nullptr)
,_kv(kv)
,_bf(0)
{}
};
template<class K,class V>
class AVLTree
{
typedef AVLTreeNode<K,V> Node;
public:
bool Insert(const pair<K, V>& kv)
{
if (_root == nullptr)
{
_root = new Node(kv);
return true;
}
Node* parent = nullptr;
Node* cur = _root;
while (cur)
{
if (cur->_kv.first > kv.first)
{
parent = cur;
cur = cur->_left;
}
else if (cur->_kv.first < kv.first)
{
parent = cur;
cur = cur->_right;
}
else
{
return false;
}
}
cur = new Node(kv);
if (parent->_kv.first > kv.first)
{
parent->_left = cur;
}
else
{
parent->_right = cur;
}
cur->_parent = parent;
//更新平衡因子
while (parent)
{
if (cur == parent->_left)
{
parent->_bf--;
}
else
{
parent->_bf++;
}
//判断
if (parent->_bf == 1 || parent->_bf == -1)
{
parent = parent->_parent;
cur = cur->_parent;
}
else if (parent->_bf == 2 || parent->_bf == -2)
{
if (parent->_bf == 2 && cur->_bf == 1)
{
RotateL(parent);
}
else if (parent->_bf == -2 && cur->_bf == -1)
{
RotateR(parent);
}
else if (parent->_bf == -2 && cur->_bf == 1)
{
RotateLR(parent);
}
else if (parent->_bf == 2 && cur->_bf == -1)
{
RotateRL(parent);
}
else
{
assert("false");
}
break;
}
else if (parent->_bf == 0)
{
break;
}
else
{
assert("false");
}
}
return true;
}
void Inorder()
{
_Inorder(_root);
}
int _Height()
{
return _Height(_root);
}
bool Isbalance()
{
return _Isbalance(_root);
}
private:
bool _Isbalance(Node* root)
{
if (root == NULL)
return true;
int leftH = _Height(root->_left);
int rightH = _Height(root->_right);
if (rightH - leftH != root->_bf)
{
cout << root->_kv.first << ":" << "平衡因子异常" << endl;
return false;
}
return abs(leftH - rightH) < 2
&& _Isbalance(root->_left)
&& _Isbalance(root->_right);
}
int _Height(Node* root)
{
if (root == NULL)
return 0;
int leftH = _Height(root->_left);
int rightH = _Height(root->_right);
return leftH > rightH ? leftH + 1 : rightH + 1;
}
void _Inorder(Node* root)
{
if (root == nullptr)
return;
_Inorder(root->_left);
cout << root->_kv.first << " ";
_Inorder(root->_right);
}
void RotateL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL = subR->_left;
parent->_right = subRL;
if (subRL)
subRL->_parent = parent;
Node* pparent = parent->_parent;
subR->_left = parent;
parent->_parent = subR;
//判断双亲的双亲节点pparent
if (pparent == nullptr)
{
_root = subR;
_root->_parent = nullptr;
}
else
{
if (pparent->_left == parent)
{
pparent->_left = subR;
}
else
{
pparent->_right = subR;
}
subR->_parent = pparent;
}
parent->_bf = subR->_bf = 0;
}
void RotateR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR = subL->_right;
parent->_left = subLR;
if (subLR)
subLR->_parent = parent;
Node* pparent = parent->_parent;
subL->_right = parent;
parent->_parent = subL;
if (pparent == nullptr)
{
_root = subLR;
_root->_parent = nullptr;
}
else
{
if (pparent->_left == parent)
{
pparent->_left = subL;
}
else
{
pparent->_right = subL;
}
subL->_parent = pparent;
}
parent->_bf = subL->_bf = 0;
}
void RotateLR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR = subL->_right;
int bf = subLR->_bf;
RotateL(parent->_left);
RotateR(parent);
if (bf == 0)
{
subLR->_bf = 0;
parent->_bf = 0;
subL->_bf = 0;
}
else if (bf == -1)
{
parent->_bf = 1;
subLR->_bf = 0;
subL->_bf = 0;
}
else if (bf == 1)
{
parent->_bf = 0;
subL->_bf = -1;
subLR->_bf = 0;
}
else
{
assert(false);
}
}
void RotateRL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL = parent->_left;
int bf = subRL->_bf;
RotateR(parent->_right);
RotateL(parent);
if (bf == 0)
{
parent->_bf = 0;
subR->_bf = 0;
subRL->_bf = 0;
}
else if (bf == -1)
{
parent->_bf = 0;
subR->_bf = 1;
subRL->_bf = 0;
}
else if (bf == 1)
{
subR->_bf = 0;
parent->_bf = -1;
subRL->_bf = 0;
}
else
{
assert(false);
}
}
private:
Node* _root = nullptr;
};
void testAVLTree()
{
AVLTree<int, int> lt;
lt.Insert(make_pair(1,1));
lt.Insert(make_pair(2,2));
lt.Insert(make_pair(7,7));
lt.Insert(make_pair(3,3));
lt.Insert(make_pair(4,4));
lt.Inorder();
cout << endl;
cout<<lt._Height();
cout << endl;
cout<<lt.Isbalance();
}
}
RL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL = parent->_left;
int bf = subRL->_bf;
RotateR(parent->_right);
RotateL(parent);
if (bf == 0)
{
parent->_bf = 0;
subR->_bf = 0;
subRL->_bf = 0;
}
else if (bf == -1)
{
parent->_bf = 0;
subR->_bf = 1;
subRL->_bf = 0;
}
else if (bf == 1)
{
subR->_bf = 0;
parent->_bf = -1;
subRL->_bf = 0;
}
else
{
assert(false);
}
}
private:
Node* _root = nullptr;
};
void testAVLTree()
{
AVLTree<int, int> lt;
lt.Insert(make_pair(1,1));
lt.Insert(make_pair(2,2));
lt.Insert(make_pair(7,7));
lt.Insert(make_pair(3,3));
lt.Insert(make_pair(4,4));
lt.Inorder();
cout << endl;
cout<<lt._Height();
cout << endl;
cout<<lt.Isbalance();
}
}
扫一扫
1332
到【灌水乐园】发言
