You Are Given a Tree
Description
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.
The first line of the input contains a single integer $n$ ($2 \le n \le 100\,000$) — the number of vertices in the tree.
Then following $n - 1$ lines describe the tree, each of them contains two integers $v$, $u$ ($1 \le v, u \le n$) — endpoints of the corresponding edge.
It is guaranteed, that the given graph is a tree.
Output $n$ numbers, the $i$-th of which is the maximum possible number of paths in an $i$-valid set of paths.
Input
The first line of the input contains a single integer $n$ ($2 \le n \le 100\,000$) — the number of vertices in the tree.
Then following $n - 1$ lines describe the tree, each of them contains two integers $v$, $u$ ($1 \le v, u \le n$) — 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.
Samples
7
1 2
2 3
3 4
4 5
5 6
6 7
7
3
2
1
1
1
1
6
1 2
2 3
2 4
1 5
5 6
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:
