CF741D Arpa’s letter-marked tree and Mehrdad’s Dokhtar-kosh paths

Just in case somebody missed it: we have wonderful girls in Arpa’s land. Arpa has a rooted tree (connected acyclic graph) consisting of $ n $ vertices. The vertices are numbered $ 1 $ through $ n $ , the vertex $ 1 $ is the root. There is a letter written on each edge of this tree. Mehrdad is a fan of Dokhtar-kosh things. He call a string Dokhtar-kosh, if we can shuffle the characters in string such that it becomes palindrome. He asks Arpa, for each vertex $ v $ , what is the length of the longest simple path in subtree of $ v $ that form a Dokhtar-kosh string.

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