Decorate Apple Tree CodeForces - 1056D（樹形dp）

There is one apple tree in Arkady’s garden. It can be represented as a set of junctions connected with branches so that there is only one way to reach any junctions from any other one using branches. The junctions are enumerated from 1 to n, the junction 1 is called the root.

A subtree of a junction v is a set of junctions u such that the path from u to the root must pass through v. Note that v itself is included in a subtree of v.

A leaf is such a junction that its subtree contains exactly one junction.

The New Year is coming, so Arkady wants to decorate the tree. He will put a light bulb of some color on each leaf junction and then count the number happy junctions. A happy junction is such a junction t that all light bulbs in the subtree of t have different colors.

Arkady is interested in the following question: for each k from 1 to n, what is the minimum number of different colors needed to make the number of happy junctions be greater than or equal to k?

Input
The first line contains a single integer n (1≤n≤105) — the number of junctions in the tree.

The second line contains n−1 integers p2, p3, …, pn (1≤pi<i), where pi means there is a branch between junctions i and pi. It is guaranteed that this set of branches forms a tree.

Output
Output n integers. The i-th of them should be the minimum number of colors needed to make the number of happy junctions be at least i.

Examples
Input
3
1 1
Output
1 1 2
Input
5
1 1 3 3
Output
1 1 1 2 3
Note
In the first example for k=1 and k=2 we can use only one color: the junctions 2 and 3 will be happy. For k=3 you have to put the bulbs of different colors to make all the junctions happy.

In the second example for k=4 you can, for example, put the bulbs of color 1 in junctions 2 and 4, and a bulb of color 2 into junction 5. The happy junctions are the ones with indices 2, 3, 4 and 5 then.

題意：
1爲根節點，葉子節點需要染色。一個點稱爲好點當且僅當其所有子樹葉子顏色不同。問滿足$k$個點爲（1≤k≤n）好點時需要的顏色數。

思路：
有一種顏色的時候，只對應一個葉子的節點成爲好點。有兩種顏色的時候，對應兩個葉子的節點可以是好點，以此類推。所以可以維護每個點對應的葉子個數。

#include <cstdio>
#include <iostream>
#include <cstring>
#include <algorithm>
#include <set>
#include <queue>
#include <map>
#include <string>
#include <iostream>
#include <cmath>

using namespace std;
typedef long long ll;

const int maxn = 1e5 + 7;

int f[maxn];
vector<int>G[maxn];
int num[maxn];

void dfs(int u,int fa) {
    if(u != 1 && G[u].size() == 1) {
        f[u] = 1;
    }
    for(int i = 0;i < G[u].size();i++) {
        int v = G[u][i];
        if(v == fa) continue;
        dfs(v,u);
        f[u] += f[v];
    }
}

int main() {
    int n;scanf("%d",&n);
    for(int i = 2;i <= n;i++) {
        int x;scanf("%d",&x);
        G[i].push_back(x);
        G[x].push_back(i);
    }
    dfs(1,-1);
    for(int i = 1;i <= n;i++) {
        num[f[i]]++;
    }
    for(int i = 1;i <= n;i++) {
        num[i] += num[i - 1];
    }
    
    int cnt = 1;
    for(int i = 1;i <= n;i++) {
        if(num[cnt] >= i) {
            printf("%d ",cnt);
        } else {
            while(num[cnt] < i) {
                cnt++;
            }
            printf("%d ",cnt);
        }
    }
    return 0;
}