You Are Given a Tree

题面翻译

有一棵 $n$ 个节点的树。

其中一个简单路径的集合被称为 $k$ 合法当且仅当：

树的每个节点至多属于其中一条路径，且每条路径恰好包含 $k$ 个点。

对于 $k\in [1,n]$，求出 $k$ 合法路径集合的最多路径数 即：设 $k$ 合法路径集合为 $S$，求最大的 $|S|$。

$n \leq 10^5$。

题目描述

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.

输入格式

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.

输出格式

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

样例 #1

样例输入 #1

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

样例输出 #1

7
3
2
1
1
1
1

样例 #2

样例输入 #2

6
1 2
2 3
2 4
1 5
5 6

样例输出 #2

6
2
2
1
1
0

提示

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