Description:
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 Format:
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 Format:
Output a single line containing $$$n$$$ space separated integers $$$a_i$$$, where $$$a_i$$$ is the maximum size of an almost-$$$i$$$-uniform set.
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 $$$\{2, 3, 5, 6\}$$$.