描述
我们可以把由“0”和“1”组成的字符串分为三类：全“0”串称为B串，全“1”串称为I串，既含“0”又含“1”的串则称为F串。
FBI树是一种二叉树1，它的结点类型也包括F结点，B结点和I结点三种。由一个长度为2^N的“01”串S可以构造出一棵FBI树T，递归的构造方法如下：
- T的根结点为R，其类型与串S的类型相同；
- 若串S的长度大于1，将串S从中间分开，分为等长的左右子串S1和S2；由左子串S1构造R的左子树T1，由右子串S2构造R的右子树T2。
现在给定一个长度为2^N的“01”串，请用上述构造方法构造出一棵FBI树，并输出它的后序遍历2序列。
格式
输入格式
输入的第一行是一个整数N(0<=N<=10)，第二行是一个长度为2^N的“01”串。
输出格式
输出包括一行，这一行只包含一个字符串，即FBI树的后序遍历序列。
样例1
样例输入1
3
10001011
样例输出1
IBFBBBFIBFIIIFF
限制
每个测试点1s
来源
NOIP2004普及组第三题
<br/ >
<nr/ >
解析：依据题意构造出这棵二叉树，然后后序遍历即可。
#include <iostream>
#include <string>
using namespace std;
struct Node{
char val;
Node *l = NULL, *r = NULL;
};
string s;
Node* build(const string& s)
{
bool flag0 = false, flag1 = false;
size_t len = s.length();
for(size_t i = 0; i < len; ++i){
if(s[i] == '0')
flag0 = true;
else
flag1 = true;
if(flag0 && flag1)
break;
}
char ch;
if(flag0 && flag1)
ch = 'F';
else if(flag0)
ch = 'B';
else
ch = 'I';
Node *root = new Node;
root->val = ch;
if(len > 1){
if(len&1){
string left = s.substr(0, len/2);
string right = s.substr(len/2+1);
root->l = build(left);
root->r = build(right);
}
else{
string left = s.substr(0, len/2);
string right = s.substr(len/2);
root->l = build(left);
root->r = build(right);
}
}
return root;
}
void post_order(Node* root)
{
if(root != NULL){
post_order(root->l);
post_order(root->r);
cout<<root->val;
}
}
void destroy(Node* root)
{
if(root->l != NULL)
destroy(root->l);
if(root->r != NULL)
destroy(root->r);
delete root;
}
int main()
{
int n;
cin>>n>>s;
Node *root = build(s);
post_order(root);
destroy(root);
return 0;
}
<br/ >
<br/ >
顺便给出先序和中序遍历:
void pre_order(Node* root)
{
if(root != NULL){
cout<<root->val;
pre_order(root->l);
pre_order(root->r);
}
}
void in_order(Node* root)
{
if(root != NULL){
in_order(root->l);
cout<<root->val;
in_order(root->r);
}
}
<br/ >
以及它们的非递归实现：
#include <iostream>
#include <string>
using namespace std;
struct Node{
char val;
Node *l = NULL, *r = NULL, *p = NULL; //增加了一个字段p，用于记录他的父结点
};
string s;
Node* build(const string& s, Node* p) //建树也做相应的更改
{
bool flag0 = false, flag1 = false;
size_t len = s.length();
for(size_t i = 0; i < len; ++i){
if(s[i] == '0')
flag0 = true;
else
flag1 = true;
if(flag0 && flag1)
break;
}
char ch;
if(flag0 && flag1)
ch = 'F';
else if(flag0)
ch = 'B';
else
ch = 'I';
Node *root = new Node;
root->val = ch;
root->p = p;
if(len > 1){
if(len&1){
string left = s.substr(0, len/2);
string right = s.substr(len/2+1);
root->l = build(left, root);
root->r = build(right, root);
}
else{
string left = s.substr(0, len/2);
string right = s.substr(len/2);
root->l = build(left, root);
root->r = build(right, root);
}
}
return root;
}
void pre_order2(Node* root)
{
Node *pre = NULL;
Node *node = root;
while(node != NULL){
if(pre == node->p){
cout<<node->val;
if(node->l != NULL){
pre = node;
node = node->l;
}
else{
if(node->r != NULL){
pre = node;
node = node->r;
}
else{
pre = node;
node = node->p;
}
}
}
else if(pre == node->l){
if(node->r != NULL){
pre = node;
node = node->r;
}
else{
pre = node;
node = node->p;
}
}
else if(pre == node->r){
pre = node;
node = node->p;
}
}
}
void in_order2(Node* root)
{
Node *pre = NULL;
Node *node = root;
while(node != NULL){
if(pre == node->p){
if(node->l != NULL){
pre = node;
node = node->l;
}
else{
cout<<node->val;
if(node->r != NULL){
pre = node;
node = node->r;
}
else{
pre = node;
node = node->p;
}
}
}
else if(pre == node->l){
cout<<node->val;
if(node->r != NULL){
pre = node;
node = node->r;
}
else{
pre = node;
node = node->p;
}
}
else if(pre == node->r){
pre = node;
node = node->p;
}
}
}
void post_order2(Node* root)
{
Node *pre = NULL;
Node *node = root;
while(node != NULL){
if(pre == node->p){
if(node->l != NULL){
pre = node;
node = node->l;
}
else{
if(node->r != NULL){
pre = node;
node = node->r;
}
else{
cout<<node->val;
pre = node;
node = node->p;
}
}
}
else if(pre == node->l){
if(node->r != NULL){
pre = node;
node = node->r;
}
else{
cout<<node->val;
pre = node;
node = node->p;
}
}
else if(pre == node->r){
cout<<node->val;
pre = node;
node = node->p;
}
}
}
void destroy(Node* root)
{
if(root->l != NULL)
destroy(root->l);
if(root->r != NULL)
destroy(root->r);
delete root;
}
int main()
{
int n;
cin>>n>>s;
Node *root = build(s, NULL);
pre_order(root);
cout<<endl;
pre_order2(root);
cout<<endl<<endl;
in_order(root);
cout<<endl;
in_order2(root);
cout<<endl<<endl;
post_order(root);
cout<<endl;
post_order2(root);
cout<<endl<<endl;
destroy(root);
return 0;
}