03-树3 Tree Traversals Again        分数 25        作者 陈越

An inorder binary tree traversal can be implemented in a non-recursive way with a stack. For example, suppose that when a 6-node binary tree (with the keys numbered from 1 to 6) is traversed, the stack operations are: push(1); push(2); push(3); pop(); pop(); push(4); pop(); pop(); push(5); push(6); pop(); pop(). Then a unique binary tree (shown in Figure 1) can be generated from this sequence of operations. Your task is to give the postorder traversal sequence of this tree.

                                                                        Figure 1

Input Specification:

Each input file contains one test case. For each case, the first line contains a positive integer N (≤30) which is the total number of nodes in a tree (and hence the nodes are numbered from 1 to N). Then 2N lines follow, each describes a stack operation in the format: "Push X" where X is the index of the node being pushed onto the stack; or "Pop" meaning to pop one node from the stack.

Output Specification:

For each test case, print the postorder traversal sequence of the corresponding tree in one line. A solution is guaranteed to exist. All the numbers must be separated by exactly one space, and there must be no extra space at the end of the line.

Sample Input:

6
Push 1
Push 2
Push 3
Pop
Pop
Push 4
Pop
Pop
Push 5
Push 6
Pop
Pop

Sample Output:

3 4 2 6 5 1

代码长度限制：16 KB        时间限制：400 ms        内存限制：64 MB

题目解析：

本质是根据一棵树的先序遍历结果和中序遍历结果，求解这棵树的后序遍历结果。

在树的遍历中，知中序和先序可求唯一后序；知中序和后序可求唯一先序；知先序和后序不可求唯一中序。

关键点1：根据输入获得先序遍历结果和中序遍历结果

关键点2：根据先序和中序遍历结果输出后序遍历结果

下面将演示建树和不建树两种情况

代码展示：

不建树的情况

// 根据先序遍历与中序遍历求后序遍历

# include<stdio.h>
# include<string.h>
# define MAXSIZE 5

void Solve(int preL, int inL, int postL, int n, int pre[], int in[], int post[]);

int main(){
	// 一共有多少个结点 
	int N;
	scanf("%d",&N);
	// 创建一个堆栈
	int Stack[N];
	int Rear = -1;
	// 分别创建前、中、后序遍历的结果数组
	int Pre[N],In[N],Post[N];
	int PreRear = -1;
	// count作为弹出元素的计数
	int num, count = 0;
	while(count!=N){
		char handle[MAXSIZE];
		scanf("%s",handle);
		if(!strcmp(handle,"Push")){
			// 接受一个数字并压入栈
			scanf("%d",&num);
			Stack[++Rear] = num; 
			// 前序遍历数组记录
			Pre[++PreRear] = num; 
		}else{
			// 出栈并在中序遍历中进行记录 
			num = Stack[Rear--];
			In[count++] = num;
		}
	} 
	
	// 有了前序与中序遍历结果，接下来通过递归函数求后序遍历结果 
	Solve(0,0,0,N,Pre,In,Post);
	int i;
	for(i=0;i<N;i++){
		if(i!=N-1)printf("%d ",Post[i]);
		else printf("%d",Post[i]);
	}
} 

void Solve(int preL, int inL, int postL, int n, int pre[], int in[], int post[]){
	if(n==0)return;
	if(n==1){
		post[postL] = pre[preL];
		return;
	}
	// 先序的第一个放在给定后序的最后一个 
	int root = pre[preL];
	post[postL+n-1] = root;
	// 找出左子树的范围 
	int i;
	for(i=0;i<n;i++){
		if(in[inL+i]==root)break;
	}
	// 左子树要处理的的结点数 
	int L = i;
	// 右子树要处理的结点数 
	int R = n-L-1;
	// 递归处理 
	Solve(preL+1, inL, postL,L,pre,in,post);
	Solve(preL+L+1,inL+L+1,postL+L,R,pre,in,post);
	return; 
}

建树的情况：

# include<stdio.h>
# include<stdlib.h>
# include<string.h>
# define MAXNODE 30
# define MAXLENGTH 5

typedef struct TreeNode* Tree;
struct TreeNode{
	int data;
	Tree left;
	Tree right;
}; 

void PostOrder(Tree root,int data);
void CreateTree(int left1,int right1,int left2,int right2,int Pre[],int In[], Tree root);

int main(){
	// 创建一个堆栈
	int Stack[MAXNODE];
	int SHead = -1; 
	// 接收结点个数 
	int N;
	scanf("%d",&N);
	getchar();
	// 分别创建一个用来记录先序和中序遍历结果的数组
	int PreArray[N],InArray[N];
	int PreHead=-1, InHead=-1;
	// 接收操作输入，因为有出入栈，因此共有2*N次操作 
	int i,num;
	char* str = (char*)malloc(sizeof(char)*MAXLENGTH);
	for(i=0;i<2*N;i++){
		scanf("%s",str);
		if(strcmp(str,"Push")==0){
			// 入栈
			scanf("%d",&num);
			getchar();
			Stack[++SHead] = num;
			// 记录先序
			PreArray[++PreHead] = num; 
		}else{
			// 出栈
			num = Stack[SHead--];
			// 记录中序
			InArray[++InHead] = num; 
		}
	}
	
	// 创建根结点
	Tree root = (Tree)malloc(sizeof(struct TreeNode));
	root->data = PreArray[0];
	root->left = NULL;
	root->right = NULL;
	// 构建树
	CreateTree(0,N-1,0,N-1,PreArray,InArray,root);
	// 后序输出遍历结果，为了格式化输出因此传入根结点的值作为判读依据 
	PostOrder(root,root->data); 
	
	return 0; 
}

// 递归后序遍历树
void PostOrder(Tree root,int data){
	if(root){
		PostOrder(root->left,data);
		PostOrder(root->right,data);
		if(root->data == data)printf("%d",root->data);
		else printf("%d ",root->data);
	}
	return;
} 

// 根据先序和后序构建一棵树；left1与right1表示先序序列的范围；left2与right2表示后序序列的范围
void CreateTree(int left1,int right1,int left2,int right2,int Pre[],int In[], Tree root){
	// 首先找到根结点在中序中的位置
	int i,index;
	for(i=left2;i<=right2;i++){
		if(root->data == In[i]){
			index = i;
			break;
		}
	} 
	// 如果根结点有左孩子 
	if(index!=left2){
		// 创建一个左孩子结点
		Tree left_son = (Tree)malloc(sizeof(struct TreeNode));
		left_son->data = Pre[left1+1];
		left_son->left = NULL;
		left_son->right = NULL;
		root->left = left_son;
		// 递归构建这个左孩子结点
		CreateTree(left1+1,index-left2+left1,left2,index-1,Pre,In,left_son); 
	}
	// 如果根结点有右孩子
	if(index!=right2){
		Tree right_son = (Tree)malloc(sizeof(struct TreeNode));
		right_son->data = Pre[index-left2+left1+1];
		right_son->left = NULL;
		right_son->right = NULL;
		root->right = right_son;
		// 递归构建这个右孩子结点
		CreateTree(index-left2+left1+1,right1,index+1,right2,Pre,In,right_son); 
	} 
	return;
} 

运行结果：

两种情况均可通过测试点