Almost Same Distance

Description:

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.

Sample Input:

5
1 2
1 3
1 4
4 5

Sample Output:

4 3 2 1 1

Sample Input:

6
1 2
1 3
1 4
4 5
4 6

Sample 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 $$$\{2, 3, 5, 6\}$$$.