最近在做些树形DP练练手
原题链接
大意就是给你一棵树，你可以断开任意数量的边，使得剩下的联通块大小乘积最大。
样例
8
1 2
1 3
2 4
2 5
3 6
3 7
6 8
输出
18
我首先想的是设\(f[i]\)表示以\(i\)为根的子树可获得的最大收益，但是会发现这样无法转移。考虑再加一维，\(f[i][j]\)表示以\(i\)的子树中，\(i\)所在的联通块大小为\(j\)的最大价值。然后我就傻了，想了半天也没想起来怎么转移，最后只好看了一眼题解。其实转移好简单的，貌似是个树上背包？考虑在\(dfs\)的过程中进行\(DP\)，每当访问完一个点\(i\)的子结点时，累加一下\(sz[i]\)，就枚举\(j\)，并且用当前子结点的\(DP\)值来更新\(f[i][j]\)。转移方程大概会长成下面这个样子：
$f[i][j]=max(f[i][j],f[i][k]*f[v][j-k])$
（理解的话，就是把之前的大小为$k$的联通块和在当前子树中大小为$j-k$的联通块拼起来）
同时，我们特别定义$f[i][0]$表示以$i$为根的子树可获得的价值，则他的转移方程比较特殊：
$f[i][0]=max(f[i][0],f[i][j]*j)$
如果到这里这道题就结束的话，代码会长成下面这样：
```cpp
#include
using namespace std;
define N 700
define ll long long
int n, eid, sz[N+5], head[N+5];
ll f[N+5][N+5];
struct Edge {
int next, to;
}e[2*N+5];
void addEdge(int u, int v) {
e[++eid].next = head[u];
e[eid].to = v;
head[u] = eid;
}
void dp(int u, int fa) {
sz[u] = 1, f[u][0] = f[u][1] = 1;
for(int i = head[u]; i; i = e[i].next) {
int v = e[i].to;
if(v == fa) continue;
dp(v, u);
sz[u] += sz[v];
for(int j = sz[u]; j >= 1; --j) { //枚举i所在的联通块大小
for(int k = min(j, sz[u]-sz[v]); k >= max(1, j-sz[v]); --k) { //枚举子树根结点所在联通块大小
f[u][j] = max(f[u][j], f[u][k]f[v][j-k]);
}
}
}
for(int i = 1; i <= sz[u]; ++i) f[u][0] = max(f[u][0], f[u][i]i);
}
int main() {
cin >> n;
for(int i = 1, x, y; i <= n-1; ++i) cin >> x >> y, addEdge(x, y), addEdge(y, x);
dp(1, 0);
cout << f[1][0] << endl;
return 0;
}
但是一交上去只有30$pts$，一看讨论区，发现还要用高精度！于是粘了个板子上去，然后就开心的$MLE$了￣▽￣。最后把$int$换成$short$就对了，无语。
粘一下$AC$代码
```cpp
#include <cstdio>
#include <iostream>
#include <cstring>
using namespace std;
#define N 700
int n, eid;
short sz[N+5], head[N+5];
struct Edge {
int next, to;
}e[2*N+5];
struct bign{ //高精类模板，网上找的
static const int maxn = 120;
short d[maxn+5];
short len;
void clean() { while(len > 1 && !d[len-1]) len--; }
bign() { memset(d, 0, sizeof(d)); len = 1; }
bign(int num) { *this = num; }
bign(char* num) { *this = num; }
bign operator = (const char* num) {
memset(d, 0, sizeof(d)); len = strlen(num);
for(int i = 0; i < len; i++) d[i] = num[len-1-i] - '0';
clean();
return *this;
}
bign operator = (int num){
char s[20]; sprintf(s, "%d", num);
*this = s;
return *this;
}
bign operator + (const bign& b) {
bign c = *this; int i;
for(i = 0; i < b.len; i++) {
c.d[i] += b.d[i];
if (c.d[i] > 9) c.d[i] %= 10, c.d[i+1]++;
}
while (c.d[i] > 9) c.d[i++] %= 10, c.d[i]++;
c.len = max(len, b.len);
if (c.d[i] && c.len <= i) c.len = i+1;
return c;
}
bign operator - (const bign& b) {
bign c = *this; int i;
for(i = 0; i < b.len; i++) {
c.d[i] -= b.d[i];
if (c.d[i] < 0) c.d[i] += 10, c.d[i+1]--;
}
while (c.d[i] < 0) c.d[i++] += 10, c.d[i]--;
c.clean();
return c;
}
bign operator * (const bign& b) const {
int i, j; bign c; c.len = len + b.len;
for(j = 0; j < b.len; j++)
for(i = 0; i < len; i++)
c.d[i+j] += d[i]*b.d[j];
for(i = 0; i < c.len-1; i++) c.d[i+1] += c.d[i]/10, c.d[i] %= 10;
c.clean();
return c;
}
bign operator / (const bign& b) {
int i, j;
bign c = *this, a = 0;
for(i = len - 1; i >= 0; i--) {
a = a*10 + d[i];
for (j = 0; j < 10; j++)
if (a < b*(j+1)) break;
c.d[i] = j;
a = a - b*j;
}
c.clean();
return c;
}
bign operator % (const bign& b) {
int i, j;
bign a = 0;
for(i = len - 1; i >= 0; i--) {
a = a*10+d[i];
for(j = 0; j < 10; j++) if (a < b*(j+1)) break;
a = a-b*j;
}
return a;
}
bign operator += (const bign& b) {
*this = *this+b;
return *this;
}
bool operator <(const bign& b) const {
if(len != b.len) return len < b.len;
for(int i = len-1; i >= 0; i--)
if(d[i] != b.d[i]) return d[i] < b.d[i];
return false;
}
bool operator >(const bign& b) const { return b < *this; }
bool operator <= (const bign& b) const { return !(b < *this); }
bool operator >= (const bign& b) const { return !(*this < b); }
bool operator != (const bign& b) const { return b < *this || *this < b; }
bool operator == (const bign& b) const { return !(b < *this) && !(b > *this); }
string str() const {
char s[maxn] = {};
for(int i = 0; i < len; i++) s[len-1-i] = d[i]+'0';
return s;
}
}f[N+5][N+5];
istream& operator >> (istream& in, bign& x) {
string s;
in >> s;
x = s.c_str();
return in;
}
ostream& operator << (ostream& out, const bign& x) {
out << x.str();
return out;
}
void addEdge(int u, int v) {
e[++eid].next = head[u];
e[eid].to = v;
head[u] = eid;
}
void dp(int u, int fa) {
sz[u] = 1, f[u][0] = f[u][1] = 1;
for(int i = head[u]; i; i = e[i].next) {
int v = e[i].to;
if(v == fa) continue;
dp(v, u);
sz[u] += sz[v];
for(int j = sz[u]; j >= 1; --j) {
for(int k = min(j, sz[u]-sz[v]); k >= max(1, j-sz[v]); --k) {
f[u][j] = max(f[u][j], f[u][k]*f[v][j-k]);
}
}
}
for(int i = 1; i <= sz[u]; ++i) f[u][0] = max(f[u][0], f[u][i]*i);
}
int main() {
cin >> n;
for(int i = 1, x, y; i <= n-1; ++i) cin >> x >> y, addEdge(x, y), addEdge(y, x);
dp(1, 0);
cout << f[1][0] << endl;
return 0;
}