CF1676G.White-Black Balanced Subtrees

You are given a rooted tree consisting of $n$ vertices numbered from $1$ to $n$ . The root is vertex $1$ . There is also a string $s$ denoting the color of each vertex: if $s_i = \texttt{B}$ , then vertex $i$ is black, and if $s_i = \texttt{W}$ , then vertex $i$ is white.

A subtree of the tree is called balanced if the number of white vertices equals the number of black vertices. Count the number of balanced subtrees.

A tree is a connected undirected graph without cycles. A rooted tree is a tree with a selected vertex, which is called the root. In this problem, all trees have root $1$ .

The tree is specified by an array of parents $a_2, \dots, a_n$ containing $n-1$ numbers: $a_i$ is the parent of the vertex with the number $i$ for all $i = 2, \dots, n$ . The parent of a vertex $u$ is a vertex that is the next vertex on a simple path from $u$ to the root.

The subtree of a vertex $u$ is the set of all vertices that pass through $u$ on a simple path to the root. For example, in the picture below, $7$ is in the subtree of $3$ because the simple path $7 \to 5 \to 3 \to 1$ passes through $3$ . Note that a vertex is included in its subtree, and the subtree of the root is the entire tree.

The picture shows the tree for $n=7$ , $a=[1,1,2,3,3,5]$ , and $s=\texttt{WBBWWBW}$ . The subtree at the vertex $3$ is balanced.

The first line of input contains an integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.

The first line of each test case contains an integer $n$ ( $2 \le n \le 4000$ ) — the number of vertices in the tree.

The second line of each test case contains $n-1$ integers $a_2, \dots, a_n$ ( $1 \le a_i < i$ ) — the parents of the vertices $2, \dots, n$ .

The third line of each test case contains a string $s$ of length $n$ consisting of the characters $\texttt{B}$ and $\texttt{W}$ — the coloring of the tree.

It is guaranteed that the sum of the values $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, output a single integer — the number of balanced subtrees.

The first test case is pictured in the statement. Only the subtrees at vertices $2$ and $3$ are balanced.

In the second test case, only the subtree at vertex $1$ is balanced.

In the third test case, only the subtrees at vertices $1$ , $3$ , $5$ , and $7$ are balanced.