D. Decorate Apple Treetime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputThere 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?InputThe first line contains a single integer n (1 \le n \le 10^5) — the number of junctions in the tree.The second line contains n - 1 integers p_2, p_3, ..., p_n (1 \le p_i < i), where p_i means there is a branch between junctions i and p_i. It is guaranteed that this set of branches forms a tree.OutputOutput 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.ExamplesInput
3
1 1
Output
1 1 2 
Input
5
1 1 3 3
Output
1 1 1 2 3 
NoteIn 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.
3
1 1

1 1 2  
['constructive algorithms', 'dfs and similar', 'dp', 'graphs', 'greedy', 'sortings', 'trees', '*1600']