Monochrome Tree

A monochrome tree is a tree in which each vertex is colored either white or black. The score of a monochrome tree is the number of unordered vertex pairs $(u, v)$ such that:

• Both vertices $u$ and $v$ are colored black.
• $u$ is an ancestor of $v$, or $v$ is an ancestor of $u$.

You are given a rooted tree of $n$ vertices whose root vertex is $1$.

For each $k$ from $0$ to $n$ (inclusive), you may color the given tree so that there are exactly $k$ black vertices and $n - k$ white vertices. Among all such colorings, let $c_k$ be the lowest score. Find $c_k$ for all $0 \leq k \leq n$.

The first line contains a single integer $n$, the number of vertices in the tree ($1 \leq n \leq 2 \times 10^5$).

The $i$-th of the following $n-1$ lines contains a single integer $p_{i+1}$ ($1 \leq p_{i+1} \leq n$, $p_{i+1} \neq i+1$), the parent of vertex $i+1$.

On the first and only line, print $n+1$ integers. The $i$-th of them is $c_{i-1}$ ($1 \leq i \leq n+1$).

Vertex $x$ is an ancestor of vertex $y$ $(x \neq y)$ if there is a sequence of vertices $\{x=a_1, a_2, \ldots a_k=y\}$ where $a_i$ is a parent of $a_{i+1}$ ($1 \leq i \leq k-1$).