【数据结构】二叉树、AVL树

08年9月入学，12年7月毕业，结束了我在软件学院愉快丰富的大学生活。此系列是对四年专业课程学习的回顾，索引参见：http://blog.csdn.net/xiaowei_cqu/article/details/7747205

 

二叉树

二叉树是每个结点最多有两个子树的有序树。通常子树的根被称作“左子树”（left subtree）和“右子树”（right subtree）。二叉树的每个结点至多只有二棵子树(不存在度大于2的结点)，二叉树的子树有左右之分，次序不能颠倒。

二叉树有几点重要的性质：

• 性质1：在二叉树的第 i 层上至多有2i-1 个结点。 (i≥1)
• 性质2：深度为 k 的二叉树上至多含2k-1 个结点（k≥1）。
• 性质3：对任何一棵二叉树，若它含有n0 个叶子结点、n2 个度为 2 的结点，则必存在关系式：n0 = n2+1。
• 性质4：具有 n 个结点的完全二叉树的深度为log2n+1
• 性质5：若对含 n 个结点的完全二叉树从上到下且从左至右进行 1 至 n 的编号，则对完全二叉树中任意一个编号为 i 的结点：
  (1) 若 i=1，则该结点是二叉树的根，无双亲，否则，编号为i/2 的结点为其双亲结点；
  (2) 若 2i>n，则该结点无左孩子，否则，编号为 2i 的结点为其左孩子结点；
  (3) 若 2i+1>n，则该结点无右孩子结点，否则，编号为2i+1 的结点为其右孩子结点。

采用链式存储结构实现二叉树

链式存储二叉树

 

1.首先我们要构造可以表示二叉树的节点的结构 Binary_node

2.构造类二叉树 Binary_tree，并编写其几个基本的成员函数：

Empty()- 检查树是否为空；clear()- 将树清空；size()- 得到树的大小；leaf_count()- 得到叶子数目；height()- 得到树高；

以及几个重要的成员函数：

Binary_tree(const Binary_tree<Entry>&original); 拷贝构造成员函数
Binary_tree &operator=(const Binary_tree<Entry>&original);重载赋值操作符
~Binary_tree();析构函数

 

3.分别编写遍历算法的成员函数

void inorder(void(*visit)(Entry &)); 中序遍历（LVR）
void preorder(void(*visit)(Entry &)); 前序遍历（VLR）
void postorder(void(*visit)(Entry &)); 后续遍历（LRV）

因为二叉树的性质，三种遍历算法我们都用递归实现，所以分别编写其递归函数

void recursive_inorder(Binary_node<Entry>*sub_root,void (*visit)(Entry &));
void recursive_preorder(Binary_node<Entry>*sub_root,void(*visit)(Entry &));
void recursive_postorder(Binary_node<Entry>*sub_root,void(*visit)(Entry &));

 

4.作为辅助，我们再编写一个print_tree的函数，用以以括号表示法输出
同样使用递归，编写递归函数void recursive_print(Binary_node<Entry>*sub_root);
几个重要的函数代码如下：

template<class Entry>
void Binary_tree<Entry>::inorder(void(*visit)(Entry &))
//Post: The tree has been traversed in inorder sequence
//Uses: The function recursive_inorder
{
	recursive_inorder(root,visit);
}

template<class Entry>
void Binary_tree<Entry>::recursive_inorder(Binary_node<Entry>*sub_root,void(*visit)(Entry &))
//Pre:  sub_root is either NULL or points to a subtree of the Binary_tree
//Post: The subtree has been traversed in inorder sequence
//Uses: The function recursive_inorder recursively
{
	if(sub_root!=NULL){
		recursive_inorder(sub_root->left,visit);
		(*visit)(sub_root->data);
		recursive_inorder(sub_root->right,visit);
	}
}

template<class Entry>
void Binary_tree<Entry>::preorder(void(*visit)(Entry &))
//Post: The tree has been traversed in preorder sequence
//Uses: The function recursive_preorder
{
	recursive_preorder(root,visit);
}

template<class Entry>
void Binary_tree<Entry>::recursive_preorder(Binary_node<Entry>*sub_root,void(*visit)(Entry &))
//Pre:  sub_root is either NULL or points to a subtree of the Binary_tree
//Post: The subtree has been traversed in preorder sequence
//Uses: The function recursive_preorder recursively
{
	if(sub_root!=NULL){
		(*visit)(sub_root->data);
		recursive_preorder(sub_root->left,visit);
		recursive_preorder(sub_root->right,visit);
	}
}

template<class Entry>
void Binary_tree<Entry>::postorder(void(*visit)(Entry &))
//Post: The tree has been traversed in postorder sequence
//Uses: The function recursive_postorder
{
	recursive_postorder(root,visit);
}

template<class Entry>
void Binary_tree<Entry>::recursive_postorder(Binary_node<Entry>*sub_root,void(*visit)(Entry &))
//Pre:  sub_root is either NULL or points to a subtree fo the Binary_tree
//Post: The subtree has been traversed in postorder sequence
//Uses: The function recursive_postorder recursively
{
	if(sub_root!=NULL){
		recursive_postorder(sub_root->left,visit);
		recursive_postorder(sub_root->right,visit);
		(*visit)(sub_root->data);
	}
}
template<class Entry>
void Binary_tree<Entry>::print_tree()
{
	recursive_print(root);
	cout<<endl;
}

template<class Entry>
void Binary_tree<Entry>::recursive_print(Binary_node<Entry>*sub_root)
{
	if(sub_root!=NULL){
		cout<<sub_root->data;
		cout<<"(";
		recursive_print(sub_root->left);
		cout<<",";
		recursive_print(sub_root->right);
		cout<<")";
	}

}
//其他函数见源码

程序结果

插入二叉树并实现中序、前序和后序遍历

 

AVL树

AVL树得名于其发明者G.M.Adelson-Velsky和E.M.Landis。AVL树是一个各结点具有平衡高度的扩展的二叉搜索树。在AVL树中，任一结点的两个子树的高度差最多为1，AVL树的高度不会超过1，AVL树既有二叉搜索树的搜索效率又可以避免二叉搜索树的最坏情况（退化树）出现。

AVL树的表示与二叉搜索树类似，其操作基本相同，但插入和删除方法除外，因为它们必须不断监控结点的左右子树的相对高度，这也正是AVL树的优势所在。

实现AVL树的相关运算

1、首先我们修改结构Binary_node，增加Balance_factor用以表示节点平衡情况

2、从二叉搜索树中派生出AVL树，编写其关键的插入和删除成员函数。

Error_code insert(const Record &new_data);
Error_code remove(const Record &old_data);

3、入和删除函数我们都用递归实现
编写递归函数：

Error_code avl_insert(Binary_node<Record>* &sub_root, 
                       const Record &new_data,bool &taller);
Error_code avl_remove(Binary_node<Record>* &sub_root,
                       const Record &target,bool &shorter);

以及几个重要的调用函数：
左旋右旋函数：

void rotate_left(Binary_node<Record>* &sub_root);
void rotate_right(Binary_node<Record>* &sub_root);

两次旋转的左右平衡函数

void right_balance(Binary_node<Record>* &sub_root);
void left_balance(Binary_node<Record>* &sub_root);

删除函数还要分别编写删除左树和删除右树的递归函数

Error_code avl_remove_right(Binary_node<Record>&sub_root,
                            const Record &target,bool &shorter);
Error_code avl_remove_left(Binary_node<Record>*&sub_root,
                            const Record &target,bool &shorter);

 

4、个重要的成员函数代码如下：

template<class Record>
Error_code AVL_tree<Record>::insert(const Record &new_data)
//Post: If the key of new_data is already in the AVL_tree, a code of duplicate_error
//      is returned. Otherwise, a code of success is returned and the Record
//      new_data is inserted into the tree in such a way that the properties of an
//      AVL tree are preserved.
{
	bool taller;
	return avl_insert(root,new_data,taller);
}

template<class Record>
Error_code AVL_tree<Record>::avl_insert(Binary_node<Record>* &sub_root, const Record &new_data,bool &taller)
//Pre:  sub_root is either NULL or points to a subtree of the AVL tree
//Post: If the key of new_data is already in the subtree, a code of duplicate_error
//      is returned. Otherwise, a code of success is returned and the Record 
//      new_data is inserted into the subtree in such a way that the properties of 
//      an AVL tree have been preserved. If the subtree is increase in height, the
//      parameter taller is set to true; otherwise it is set to false
//Uses: Methods of struct AVL_node; functions avl_insert recursively, left_balance, and right_balance
{
	Error_code result=success;
	if(sub_root==NULL){
		sub_root=new Binary_node<Record>(new_data);
		taller=true;
	}
	else if(new_data==sub_root->data){
		result=duplicate_error;
		taller=false;
	}
	else if(new_data<sub_root->data){//Insert in left subtree
		result=avl_insert(sub_root->left,new_data,taller);
		if(taller==true)
			switch(sub_root->get_balance()){//Change balance factors
			case left_higher:
				left_balance(sub_root);
				taller=false;
				break;
			case equal_height:
				sub_root->set_balance(left_higher);
				break;
			case right_higher:
				sub_root->set_balance(equal_height);
				taller=false;
				break;
		}
	}
	else{        //Insert in right subtree
		result=avl_insert(sub_root->right,new_data,taller);
		if(taller==true)
			switch(sub_root->get_balance()){
			case left_higher:
				sub_root->set_balance(equal_height);
				taller=false;
				break;
			case equal_height:
				sub_root->set_balance(right_higher);
				break;
			case right_higher:
				right_balance(sub_root);
				taller=false; //Rebalancing always shortens the tree
				break;
		}
	}
	return result;
}

template<class Record>
void AVL_tree<Record>::right_balance(Binary_node<Record>* &sub_root)
//Pre:  sub_root points to a subtree of an AVL_tree that is doubly unbalanced
//      on the right
//Post: The AVL properties have been restored to the subtree
{
	Binary_node<Record>* &right_tree=sub_root->right;
	switch(right_tree->get_balance()){
	case right_higher:
		sub_root->set_balance(equal_height);
		right_tree->set_balance(equal_height);
		rotate_left(sub_root);
		break;
	case equal_height:
		cout<<"WARNING: program error detected in right_balance "<<endl;
	case left_higher:
		Binary_node<Record>*sub_tree=right_tree->left;
		switch(sub_tree->get_balance()){
		case equal_height:
			sub_root->set_balance(equal_height);
			right_tree->set_balance(equal_height);
			break;
		case left_higher:
			sub_root->set_balance(equal_height);
			right_tree->set_balance(right_higher);
		case right_higher:
			sub_root->set_balance(left_higher);
			right_tree->set_balance(equal_height);
			break;
		}
		sub_tree->set_balance(equal_height);
		rotate_right(right_tree);
		rotate_left(sub_root);
		break;
	}
}

template<class Record>
void AVL_tree<Record>::left_balance(Binary_node<Record>* &sub_root)
{
	Binary_node<Record>* &left_tree=sub_root->left;
	switch(left_tree->get_balance()){
	case left_higher:
		sub_root->set_balance(equal_height);
		left_tree->set_balance(equal_height);
		rotate_right(sub_root);
		break;
	case equal_height:
		cout<<"WARNING: program error detected in left_balance"<<endl;
	case right_higher:
		Binary_node<Record>*sub_tree=left_tree->right;
		switch(sub_tree->get_balance()){
		case equal_height:
			sub_root->set_balance(equal_height);
			left_tree->set_balance(equal_height);
			break;
		case right_higher:
			sub_root->set_balance(equal_height);
			left_tree->set_balance(left_higher);
			break;
		case left_higher:
			sub_root->set_balance(right_higher);
			left_tree->set_balance(equal_height);
			break;
		}
		sub_tree->set_balance(equal_height);
		rotate_left(left_tree);
		rotate_right(sub_root);
		break;
	}
}

template<class Record>
void AVL_tree<Record>::rotate_left(Binary_node<Record>* &sub_root)
//Pre:  sub_root points to a subtree of the AVL_tree. This subtree has 
//      a nonempty right subtree.
//Post: sub_root is reset to point to its former right child, and the 
//      former sub_root node is the left child of the new sub_root node
{
	if(sub_root==NULL||sub_root->right==NULL)//impossible cases
		cout<<"WARNING: program error detected in rotate_left"<<endl;
	else{
		Binary_node<Record>*right_tree=sub_root->right;
		sub_root->right=right_tree->left;
		right_tree->left=sub_root;
		sub_root=right_tree;
	}
}

template<class Record>
void AVL_tree<Record>::rotate_right(Binary_node<Record>*&sub_root)
{
	if(sub_root==NULL||sub_root->left==NULL)
		cout<<"WARNING:program error in detected in rotate_right"<<endl;
	else{
		Binary_node<Record>*left_tree=sub_root->left;
		sub_root->left=left_tree->right;
		left_tree->right=sub_root;
		sub_root=left_tree;
	}
}

template<class Record>
Error_code AVL_tree<Record>::remove(const Record &old_data)
{
	bool shorter;
	return avl_remove(root,old_data,shorter);
}

template<class Record>
Error_code AVL_tree<Record>::avl_remove(Binary_node<Record>* &sub_root,
										const Record &target,bool &shorter)
{
	Binary_node<Record>*temp;
	if(sub_root==NULL)return fail;
	else if(target<sub_root->data)
		return avl_remove_left(sub_root,target,shorter);
	else if(target>sub_root->data)
		return avl_remove_right(sub_root,target,shorter);
	else if(sub_root->left==NULL){//Found target: delete current node
		temp=sub_root;       //Move right subtree up to delete node
		sub_root=sub_root->right;
		delete temp;
		shorter=true;
	}
	else if(sub_root->right==NULL){
		temp=sub_root;   //Move left subtree up to delete node
		sub_root=sub_root->left;
		delete temp;
		shorter=true;
	}
	else if(sub_root->get_balance()==left_higher){
		//Neither subtree is empty; delete from the taller
		temp=sub_root->left;//Find predecessor of target and delete if from left tree
		while(temp->right!=NULL)temp=temp->right;
		sub_root->data=temp->data;
		avl_remove_left(sub_root,temp->data,shorter);
	}
	else{
		temp=sub_root->right;
		while(temp->left!=NULL)temp=temp->left;
		sub_root->data=temp->data;
		avl_remove_right(sub_root,temp->data,shorter);
	}
	return success;
}

template<class Record>
Error_code AVL_tree<Record>::avl_remove_right(Binary_node<Record>
		   *&sub_root,const Record &target,bool &shorter)
{
	Error_code result=avl_remove(sub_root->right,target,shorter);
	if(shorter==true)switch(sub_root->get_balance()){
		case equal_height:
			sub_root->set_balance(left_higher);
			shorter=false;
			break;
		case right_higher:
			sub_root->set_balance(equal_height);
			break;
		case left_higher:
			Binary_node<Record>*temp=sub_root->left;
			switch(temp->get_balance()){
			case equal_height:
				temp->set_balance(right_higher);
				rotate_right(sub_root);
				shorter=false;
				break;
			case left_higher:
				sub_root->set_balance(equal_height);
				temp->set_balance(equal_height);
				rotate_right(sub_root);
				break;
			case right_higher:
				Binary_node<Record>*temp_right=temp->right;
				switch(temp_right->get_balance()){
				case equal_height:
					sub_root->set_balance(equal_height);
					temp->set_balance(equal_height);
					break;
				case left_higher:
					sub_root->set_balance(right_higher);
					temp->set_balance(equal_height);
					break;
				case right_higher:
					sub_root->set_balance(equal_height);
					temp->set_balance(left_higher);
					break;
				}
				temp_right->set_balance(equal_height);
				rotate_left(sub_root->left);
				rotate_right(sub_root);
				break;
			}
	}
	return result;
}

template<class Record>
Error_code AVL_tree<Record>::avl_remove_left(Binary_node<Record>
		   *&sub_root,const Record &target,bool &shorter)
{
	Error_code result=avl_remove(sub_root->left,target,shorter);
	if(shorter==true)
		switch(sub_root->get_balance()){
		case equal_height:
			sub_root->set_balance(right_higher);
			shorter=false;
			break;
		case left_higher:
			sub_root->set_balance(equal_height);
			break;
		case right_higher:
			Binary_node<Record>*temp=sub_root->right;
			switch(temp->get_balance()){
			case equal_height:
				temp->set_balance(left_higher);
				rotate_right(sub_root);
				shorter=false;
				break;
			case right_higher:
				sub_root->set_balance(equal_height);
				temp->set_balance(equal_height);
				rotate_left(sub_root);
				break;
			case left_higher:
				Binary_node<Record>*temp_left=temp->left;
				switch(temp_left->get_balance()){
				case equal_height:
					sub_root->set_balance(equal_height);
					temp->set_balance(equal_height);
					break;
				case right_higher:
					sub_root->set_balance(left_higher);
					temp->set_balance(equal_height);
					break;
				case left_higher:
					sub_root->set_balance(equal_height);
					temp->set_balance(right_higher);
					break;
				}
				temp_left->set_balance(equal_height);
				rotate_right(sub_root->right);
				rotate_left(sub_root);
				break;
			}
	}
	return result;
}

实验结果

实现如下功能：1）由{4,9,0,1,8,6,3,5,2,7}建AVL树B，并以括号表示法输出；2）删除B中关键字为8和2的结点，输出结果。

 

分析总结

 

采用链式存储结构实现二叉树

1. 二叉树在插入的时候通过判断待插入数据与根节点数据的大小，若小于根数据，插入左树，反之插入右树（我们规定不许有重复元素）；二叉树的存储结构常用于搜索等。但搜索的效率常依赖于树的结构。而树的结构与元素插入顺序有很大关系，用上述方法插入时若插入的元素是有序的，则得到的树与队列几乎没有区别，也起不到优化搜索的目的。
2. 二叉树的遍历算法最为重要的一点是递归，递归使我们不必关心于具体的遍历每个节点的顺序，而只是将其看做三个部分，即左子树，根节点，右子树，具体子树的遍历仍是调用其递归函数。
   3.在打印树的时候，我写的并不完美，因为对于叶子节点，理想应该不再打印括号，但我通过判断根节点不为空而调用递归函数，即只要节点有元素就会输出（，），还是表达出了树的意思，但并没有想到怎样达到简洁的效果。

实现AVL树的相关运算

1. AVL树每插入，删除一个节点就会判断树的高度并改变相应节点的平衡因素，每当有节点不再满足AVL树的性质，立即通过旋转得到AVL树
2. 旋转的函数及其巧妙。而插入和删除元素的函数要考虑的情况非常多，一种情况没有考虑到就可能达不到我们想要的效果，函数的编写需要极大的耐心和对编程语句的熟练掌握。很多地方我是参考书上的，别人的代码我可以理解，但我自己写可能还是会漏掉很多情况。
   

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