平衡二叉樹(AVL)的插入和刪除詳解(上)：http://blog.csdn.net/sysu_arui/article/details/7897017
平衡二叉樹(AVL)的插入和刪除詳解(下)：http://blog.csdn.net/sysu_arui/article/details/7906303
前面詳細介紹了平衡二叉樹的插入和刪除的實現，前面實現中結點使用的是一個值爲-1、0、1的平衡因子bf，這裏給出的實現其結點使用的域爲高度height
空樹高度爲-1，單節點樹高度爲0，一棵樹的高度等於其左子樹高度和右子樹高度中的較大者加1，通過比較左右子樹的高度，我們很容易求出平衡因子，但是我們不顯示的求平衡因子，因爲平衡因子也就是左右子樹的高度差，使用樹的高度是便於平衡因子的維護。
前面講解中，在插入和刪除後，平衡因子變化很複雜，而下面給出的代碼在插入和刪除後只需要自頂向上求樹的高度，不需要複雜的計算平衡因子，自己比較代碼會有體會。
代碼如下：
#include <cstdlib>
#include <iostream>
using namespace std;
template <typename Comparable>
class AVLTree
{
public:
AVLTree() : root(NULL){}
AVLTree(const AVLTree & rhs) : root(NULL)
{
*this = rhs;
}
~AVLTree()
{
makeEmpty();
}
/**
* Find the smallest item in the tree.
*/
const Comparable & findMin() const
{
if(!isEmpty())
{
return findMin(root)->element;
}
}
/**
* Find the largest item in the tree.
*/
const Comparable & findMax() const
{
if(!isEmpty())
{
return findMax(root)->element;
}
}
/**
* Returns true if x is found in the tree.
*/
bool contains(const Comparable & x) const
{
return contains(x, root);
}
/**
* Test if the tree is logically empty.
* Return true if empty, false otherwise.
*/
bool isEmpty() const
{
return root == NULL;
}
/**
* Print the tree contents in sorted order.
*/
void printTree() const
{
if(isEmpty())
{
cout << "Empty tree" << endl;
}
else
{
printTree(root);
}
}
/**
* Make the tree logically empty.
*/
void makeEmpty()
{
makeEmpty(root);
}
/**
* Insert x into the tree; duplicates are ignored.
*/
void insert(const Comparable & x )
{
insert(x,root);
}
/**
* Remove x from the tree. Nothing is done if x is not found.
*/
void remove(const Comparable & x)
{
remove(x,root);
}
/**
* Deep copy.
*/
const AVLTree & operator=(const AVLTree & rhs)
{
if( this != &rhs )
{
makeEmpty( );
root = clone(rhs.root);
}
return *this;
}
private:
struct AVLNode
{
Comparable element;
AVLNode* left;
AVLNode* right;
int height;
AVLNode(const Comparable & theElement, AVLNode *lt,
AVLNode *rt, int h = 0)
: element(theElement), left(lt), right(rt), height(h) {}
};
AVLNode *root;
/**
* Internal method to insert into a subtree.
* x is the item to insert.
* t is the node that roots the subtree.
* Set the new root of the subtree.
*/
void insert(const Comparable & x, AVLNode * & t)
{
if(t == NULL)
{
t = new AVLNode(x, NULL, NULL);
}
else if(x < t->element)
{
insert(x, t->left);
if(height(t->left) - height(t->right) == 2)
leftBalance(t);
// if(x < t->left->element)
// rotateWithLeftChild(t);//LL rotate
// else
// doubleWithLeftChild(t);//LR rotate
}
else if(t->element < x)
{
insert(x, t->right);
if( height(t->right) - height(t->left) == 2)
rightBalance(t);
// if(t->right->element < x)
// rotateWithRightChild(t);//RR rotate
// else
// doubleWithRightChild(t);//RL rotate
}
else
; // Duplicate; do nothing
t->height = max(height(t->left), height(t->right)) + 1;
}
/**
* Internal method to remove in a subtree.
* x is the item to remove.
* t is the node that roots the subtree.
* Set the new root of the subtree.
*/
void remove(const Comparable & x, AVLNode * & t)
{
if(t == NULL) //no such element
{
return;
}
else if(x < t->element) //find in left subtree
{
remove(x, t->left);
if(height(t->right) - height(t->left) == 2)
rightBalance(t);
}
else if(t->element < x) //find in right subtree
{
remove(x, t->right);
if( height(t->left) - height(t->right) == 2)
leftBalance(t);
}
else //delete node *t，
{
if(t->left == NULL)
{
AVLNode* q = t;
t = t->right;
delete q;
}
else if(t->right == NULL)
{
AVLNode* q = t;
t = t->left;
delete q;
}
else
{
t->element = findMax(t->left)->element;
remove(t->element,t->left);
//t->element = findMin(t->right)->element;
//remove(t->element,t->right);
}
}
if(t)
t->height = max(height(t->left), height(t->right)) + 1;
}
/**
* Internal method to find the smallest item in a subtree t.
* Return node containing the smallest item.
*/
AVLNode * findMin(AVLNode *t) const
{
if(t == NULL)
return NULL;
if(t->left == NULL)
return t;
return findMin(t->left);
}
/**
* Internal method to find the largest item in a subtree t.
* Return node containing the largest item.
*/
AVLNode * findMax(AVLNode *t) const
{
if(t != NULL)
while(t->right != NULL)
t = t->right;
return t;
}
/**
* Internal method to test if an item is in a subtree.
* x is item to search for.
* t is the node that roots the tree.
*/
bool contains(const Comparable & x, AVLNode *t) const
{
if( t == NULL )
return false;
else if( x < t->element )
return contains( x, t->left );
else if( t->element < x )
return contains( x, t->right );
else
return true; // Match
}
/****** NONRECURSIVE VERSION*************************
bool contains( const Comparable & x, AVLNode *t ) const
{
while( t != NULL )
if( x < t->element )
t = t->left;
else if( t->element < x )
t = t->right;
else
return true; // Match
return false; // No match
}
*****************************************************/
/**
* Internal method to make subtree empty.
*/
void makeEmpty(AVLNode * & t)
{
if(t != NULL)
{
makeEmpty(t->left);
makeEmpty(t->right);
delete t;
}
t = NULL;
}
void preOrder(AVLNode *t)const
{
if(t)
{
cout<<t->element<<" ";
preOrder(t->left);
preOrder(t->right);
}
}
void inOrder(AVLNode *t)const
{
if(t)
{
inOrder(t->left);
cout<<t->element<<" ";
inOrder(t->right);
}
}
/**
* Internal method to print a subtree rooted at t in sorted order.
*/
void printTree(AVLNode *t) const
{
if(t)
{
cout<<"preOrder: "<<endl;
preOrder(t);
cout<<endl;
cout<<"inOrder: "<<endl;
inOrder(t);
cout<<endl;
}
}
/**
* Internal method to clone subtree.
*/
AVLNode * clone(AVLNode *t) const
{
if( t == NULL )
return NULL;
else
return new AVLNode(t->element, clone(t->left), clone(t->right), t->height );
}
// Avl manipulations
/**
* Return the height of node t or -1 if NULL.
* since the height of a tree with single node is 0
*/
int height(AVLNode *t) const
{
return t == NULL ? -1 : t->height;
}
int max(int lhs, int rhs) const
{
return lhs > rhs ? lhs : rhs;
}
/**
* Rotate binary tree node with left child.
* For AVL trees, this is a single rotation for case 1-LL.
* Update heights, then set new root.
*/
void rotateWithLeftChild(AVLNode * & k2)
{
AVLNode *k1 = k2->left;
k2->left = k1->right;
k1->right = k2;
k2->height = max(height(k2->left), height(k2->right)) + 1;
k1->height = max( height(k1->left), k2->height) + 1;
k2 = k1;
}
/**
* Rotate binary tree node with right child.
* For AVL trees, this is a single rotation for case 4-RR.
* Update heights, then set new root.
*/
void rotateWithRightChild(AVLNode * & k1)
{
AVLNode *k2 = k1->right;
k1->right = k2->left;
k2->left = k1;
k1->height = max(height(k1->left), height(k1->right)) + 1;
k2->height = max(height(k2->right), k1->height) + 1;
k1 = k2;
}
/**
* Double rotate binary tree node: first left child.
* with its right child; then node k3 with new left child.
* For AVL trees, this is a double rotation for case 2-LR.
* Update heights, then set new root.
*/
void doubleWithLeftChild(AVLNode * & k3)
{
rotateWithRightChild( k3->left );
rotateWithLeftChild( k3 );
}
/**
* Double rotate binary tree node: first right child.
* with its left child; then node k1 with new right child.
* For AVL trees, this is a double rotation for case 3-RL.
* Update heights, then set new root.
*/
void doubleWithRightChild(AVLNode * & k1)
{
rotateWithLeftChild(k1->right);
rotateWithRightChild(k1);
}
/**
* left balance the subtree with root t
* this method can use for both insert and delete
*/
void leftBalance(AVLNode *& t)
{
AVLNode* lc = t->left;
if(height(lc->left) - height(lc->right) == -1)
{
doubleWithLeftChild(t); //LR rotate
}
else
{
rotateWithLeftChild(t); //LL rotate
}
}
/**
* right balance the subtree with root t
* this method can use for both insert and delete
*/
void rightBalance(AVLNode *& t)
{
AVLNode* rc = t->right;
if(height(rc->left) - height(rc->right) == 1)
{
doubleWithRightChild(t); //RL rotate
}
else
{
rotateWithRightChild(t); //RR rotate
}
}
};
int main(int argc, char *argv[])
{
const int N = 20;
AVLTree<int> t;
//insert
for(int i=0; i<N; i++)
{
t.insert(i);
}
cout<<"after insert:"<<endl;
t.printTree();
cout<<endl<<endl;
//remove
for(int i=1; i<N; i += 2)
{
t.remove(i);
}
cout<<"after remove:"<<endl;
t.printTree();
cout<<endl<<endl;
t.makeEmpty();
system("PAUSE");
return EXIT_SUCCESS;
}
注：
(1)實現中大部分代碼來自參考資料[1]，但是書中沒有給出刪除的具體實現，本人實現了一個，僅供參考。
(2)添加了左平衡和右平衡處理函數，是爲了處理刪除操作，但是該函數對插入操作也適用，看上面插入函數被註釋的代碼(被leftBalance函數和rightBalance函數替代)
參考資料：
[1]Data Structures and Algorithm Analysis in C++(third editon) (數據結構與算法分析C++描述，第3版 )
[2][嚴蔚敏《數據結構(C語言版)》
[3]算法導論（第2版）P177
