You Are Given a Tree

A tree is an undirected graph with exactly one simple path between each pair of vertices. We call a set of simple paths k-valid if each vertex of the tree belongs to no more than one of these paths (including endpoints) and each path consists of exactly k vertices.

You are given a tree with n vertices. For each k from 1 to n inclusive find what is the maximum possible size of a k-valid set of simple paths.

Input

The first line of the input contains a single integer n (2 ≤ n ≤ 100 000) — the number of vertices in the tree.

Then following n - 1 lines describe the tree, each of them contains two integers v, u (1 ≤ v, u ≤ n) — endpoints of the corresponding edge.

It is guaranteed, that the given graph is a tree.

Output

Output n numbers, the i-th of which is the maximum possible number of paths in an i-valid set of paths.

Examples

Input

7 1 2 2 3 3 4 4 5 5 6 6 7

Output

7 3 2 1 1 1 1

Input

6 1 2 2 3 2 4 1 5 5 6

Output

6 2 2 1 1 0

Note

One way to achieve the optimal number of paths for the second sample is illustrated in the following picture:

inputFormat

Input

The first line of the input contains a single integer n (2 ≤ n ≤ 100 000) — the number of vertices in the tree.

Then following n - 1 lines describe the tree, each of them contains two integers v, u (1 ≤ v, u ≤ n) — endpoints of the corresponding edge.

It is guaranteed, that the given graph is a tree.

outputFormat

Output

Output n numbers, the i-th of which is the maximum possible number of paths in an i-valid set of paths.

Examples

Input

7 1 2 2 3 3 4 4 5 5 6 6 7

Output

7 3 2 1 1 1 1

Input

6 1 2 2 3 2 4 1 5 5 6

Output

6 2 2 1 1 0

Note

One way to achieve the optimal number of paths for the second sample is illustrated in the following picture:

样例

6
1 2
2 3
2 4
1 5
5 6

6
2
2
1
1
0

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