P16135 [ICPC 2018 NAIPC] Red Black Tree

You are given a rooted tree with $n$ nodes. The nodes are numbered $1\dots n$. The root is node 1, and $m$ of the nodes are colored red, the rest are black. You would like to choose a subset of nodes such that there is no node in your subset which is an ancestor of any other node in your subset. For example, if A is the parent of B and B is the parent of C, then you could have at most one of A, B or C in your subset. In addition, you would like exactly $k$ of your chosen nodes to be red. If exactly $m$ of the nodes are red, then for all $k = 0\dots m$, figure out how many ways you can choose subsets with $k$ red nodes, and no node is an ancestor of any other node.

Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each test case will begin with a line with two integers $n$ ($1 \leq n \leq 2 \times 10^5$) and $m$ ($0 \leq m \leq \min(10^3, n)$), where $n$ is the number of nodes in the tree, and $m$ is the number of nodes which are red. The nodes are numbered $1..n$. Each of the next $n - 1$ lines will contain a single integer $p$ ($1 \leq p \leq n$), which is the number of the parent of this node. The nodes are listed in order, starting with node 2, then node 3, and so on. Node 1 is skipped, since it is the root. It is guaranteed that the nodes form a single tree, with a single root at node 1 and no cycles. Each of the next $m$ lines will contain single integer $r$ ($1 \leq r \leq n$). These are the numbers of the red nodes. No value of $r$ will be repeated.

Output $m + 1$ lines, corresponding to the number of subsets satisfying the given criteria with a number of red nodes equal to $k = 0\dots m$, in that order. Output this number modulo $10^9 + 7$.