CF914E 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.

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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 $