[NOIP2003]加分二叉树
中序遍历有一个特点：序列[1, n]是一个中序遍历，i 是根节点，则[1, i - 1]是他的左子树的中序遍历，[i + 1, n]为右子树的中序遍历。所以就想到了区间dp，对于[i, j]枚举根节点k，则dp[i][j] = max(dp[i][k - 1] * dp[k + 1][j] + a[k])。初始化dp[i][i] = a[i]，又因为空节点为1，所以dp[i][i - 1] = 1。
前序遍历的话就再开一个二维数组f[i][j]存这个区间所属哪一个节点，输出的时候跟区间dp很像，具体看代码吧。
1 #include<cstdio> 2 #include<iostream> 3 #include<cmath> 4 #include<algorithm> 5 #include<cstring> 6 #include<cstdlib> 7 #include<cctype> 8 #include<vector> 9 #include<stack> 10 #include<queue> 11 using namespace std; 12 #define enter puts("") 13 #define space putchar(' ') 14 #define Mem(a, x) memset(a, x, sizeof(a)) 15 #define rg register 16 typedef long long ll; 17 typedef double db; 18 const int INF = 0x3f3f3f3f; 19 const db eps = 1e-8; 20 const int maxn = 35; 21 inline ll read() 22 { 23 ll ans = 0; 24 char ch = getchar(), last = ' '; 25 while(!isdigit(ch)) {last = ch; ch = getchar();} 26 while(isdigit(ch)) {ans = ans * 10 + ch - '0'; ch = getchar();} 27 if(last == '-') ans = -ans; 28 return ans; 29 } 30 inline void write(ll x) 31 { 32 if(x < 0) x = -x, putchar('-'); 33 if(x >= 10) write(x / 10); 34 putchar(x % 10 + '0'); 35 } 36 37 int n, a[maxn]; 38 ll dp[maxn][maxn]; 39 int f[maxn][maxn]; 40 41 void print(int L, int R) 42 { 43 if(R < L) return; 44 if(L == R) {write(L), space; return;} 45 write(f[L][R]), space; 46 print(L, f[L][R] - 1); 47 print(f[L][R] + 1, R); 48 } 49 50 int main() 51 { 52 n = read(); 53 for(int i = 1; i <= n; ++i) a[i] = read(); 54 for(int i = 1; i <= n; ++i) dp[i][i] = a[i], dp[i][i - 1] = 1; 55 for(int L = 2; L <= n; ++L) 56 for(int i = 1; i + L - 1 <= n; ++i) 57 { 58 int j = i + L - 1; 59 for(int k = i; k <= j; ++k) 60 { 61 ll Max = dp[i][k - 1] * dp[k + 1][j] + a[k]; 62 if(Max > dp[i][j]) dp[i][j] = Max, f[i][j] = k; 63 } 64 } 65 write(dp[1][n]); enter; 66 print(1, n); 67 return 0; 68 }
