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F. Almost Same Distance
time limit per test
5 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Let $$$G$$$ be a simple graph. Let $$$W$$$ be a non-empty subset of vertices. Then $$$W$$$ is almost-$$$k$$$-uniform if for each pair of distinct vertices $$$u,v \in W$$$ the distance between $$$u$$$ and $$$v$$$ is either $$$k$$$ or $$$k+1$$$.

You are given a tree on $$$n$$$ vertices. For each $$$i$$$ between $$$1$$$ and $$$n$$$, find the maximum size of an almost-$$$i$$$-uniform set.

Input

The first line contains a single integer $$$n$$$ ($$$2 \leq n \leq 5 \cdot 10^5$$$) – the number of vertices of the tree.

Then $$$n-1$$$ lines follows, the $$$i$$$-th of which consisting of two space separated integers $$$u_i$$$, $$$v_i$$$ ($$$1 \leq u_i, v_i \leq n$$$) meaning that there is an edge between vertices $$$u_i$$$ and $$$v_i$$$.

It is guaranteed that the given graph is tree.

Output

Output a single line containing $$$n$$$ space separated integers $$$a_i$$$, where $$$a_i$$$ is the maximum size of an almost-$$$i$$$-uniform set.

Examples
Input
5
1 2
1 3
1 4
4 5
Output
4 3 2 1 1
Input
6
1 2
1 3
1 4
4 5
4 6
Output
4 4 2 1 1 1
Note

Consider the first example.

  • The only maximum almost-$$$1$$$-uniform set is $$$\{1, 2, 3, 4\}$$$.
  • One of the maximum almost-$$$2$$$-uniform sets is or $$$\{2, 3, 5\}$$$, another one is $$$\{2, 3, 4\}$$$.
  • A maximum almost-$$$3$$$-uniform set is any pair of vertices on distance $$$3$$$.
  • Any single vertex is an almost-$$$k$$$-uniform set for $$$k \geq 1$$$.

In the second sample there is an almost-$$$2$$$-uniform set of size $$$4$$$, and that is $$$\{2, 3, 5, 6\}$$$.