You Are Given a Tree

题意翻译

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.

输入输出格式

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.

输入输出样例

输入样例 #1

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

输出样例 #1

7
3
2
1
1
1
1

输入样例 #2

6
1 2
2 3
2 4
1 5
5 6

输出样例 #2

6
2
2
1
1
0

说明

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