Red Black Tree

You are given a rooted tree with $n$ nodes, numbered $1$ to $n$. The root is node $1$. Exactly $m$ of the nodes are red, and the rest are black.

Choose a subset of nodes such that no chosen node is an ancestor of another chosen node. For example, if $A$ is the parent of $B$ and $B$ is the parent of $C$, then at most one of $A$, $B$, and $C$ can be in the subset. You want exactly $k$ of the chosen nodes to be red.

For each $k = 0, \ldots, m$, count the subsets that contain exactly $k$ red nodes and satisfy the condition above.

The first line contains two integers $n$ and $m$ ($1 \le n \le 2 \times 10^5$, $0 \le m \le \min(10^3, n)$). Each of the next $n-1$ lines contains one integer $p$, the parent of node $2$, then node $3$, and so on. Each of the next $m$ lines contains one integer $r$, the number of a red node. No value of $r$ repeats.

Print $m+1$ lines. Line $k+1$ contains the number of valid subsets with exactly $k$ red nodes, modulo $10^9 + 7$.