D. You Are Given a Tree
time limit per test
7 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

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 $n$ ($2 \le n \le 100\,000$) — 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.

Examples
Input
71 22 33 44 55 66 7
Output
7321111
Input
61 22 32 41 55 6
Output
622110
Note

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