A Wide, Wide Graph
Description
You are given a tree (a connected graph without cycles) with $n$ vertices.
Consider a fixed integer $k$. Then, the graph $G_k$ is an undirected graph with $n$ vertices, where an edge between vertices $u$ and $v$ exists if and only if the distance between vertices $u$ and $v$ in the given tree is at least $k$.
For each $k$ from $1$ to $n$, print the number of connected components in the graph $G_k$.
The first line contains the integer $n$ ($2 \le n \le 10^5$) — the number of vertices in the graph.
Each of the next $n-1$ lines contains two integers $u$ and $v$ ($1 \le u, v \le n$), denoting an edge between vertices $u$ and $v$ in the tree. It is guaranteed that these edges form a valid tree.
Output $n$ integers: the number of connected components in the graph $G_k$ for each $k$ from $1$ to $n$.
Input
The first line contains the integer $n$ ($2 \le n \le 10^5$) — the number of vertices in the graph.
Each of the next $n-1$ lines contains two integers $u$ and $v$ ($1 \le u, v \le n$), denoting an edge between vertices $u$ and $v$ in the tree. It is guaranteed that these edges form a valid tree.
Output
Output $n$ integers: the number of connected components in the graph $G_k$ for each $k$ from $1$ to $n$.
6
1 2
1 3
2 4
2 5
3 6
5
1 2
2 3
3 4
3 5
1 1 2 4 6 6
1 1 3 5 5
Note
In the first example: If $k=1$, the graph has an edge between each pair of vertices, so it has one component. If $k=4$, the graph has only edges $4 \leftrightarrow 6$ and $5 \leftrightarrow 6$, so the graph has $4$ components.
In the second example: when $k=1$ or $k=2$ the graph has one component. When $k=3$ the graph $G_k$ splits into $3$ components: one component has vertices $1$, $4$ and $5$, and two more components contain one vertex each. When $k=4$ or $k=5$ each vertex is a separate component.