A11510. Palindromes in a Tree

You are given a tree (a connected acyclic undirected graph) of $n$ vertices. Vertices are numbered from $1$ to $n$ and each vertex is assigned a character from a to t.

A path in the tree is said to be palindromic if at least one permutation of the labels in the path is a palindrome.

For each vertex, output the number of palindromic paths passing through it.

Note: The path from vertex $u$ to vertex $v$ is considered to be the same as the path from vertex $v$ to vertex $u$ , and this path will be counted only once for each of the vertices it passes through.

The first line contains an integer $n$ ( $2<=n<=2·10^{5}$ ) — the number of vertices in the tree.

The next $n-1$ lines each contain two integers $u$ and $v$ ( $1<=u,v<=n,u≠v$ ) denoting an edge connecting vertex $u$ and vertex $v$ . It is guaranteed that the given graph is a tree.

The next line contains a string consisting of $n$ lowercase characters from a to t where the $i$ -th ( $1<=i<=n$ ) character is the label of vertex $i$ in the tree.

Print $n$ integers in a single line, the $i$ -th of which is the number of palindromic paths passing through vertex $i$ in the tree.

In the first sample case, the following paths are palindromic:

$2-3-4$

$2-3-5$

$4-3-5$

Additionally, all paths containing only one vertex are palindromic. Listed below are a few paths in the first sample that are not palindromic:

$1-2-3$

$1-2-3-4$

$1-2-3-5$