403514: GYM101189 D Arpa’s letter-marked tree and Mehrdad’s Dokhtar-kosh paths(Hard)

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 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.

The first line contains integer n (1  ≤  n  ≤  5·10⁵) — the number of vertices in the tree.

(n  -  1) lines follow, the i-th of them contain an integer p_i and a letter c_i (1  ≤  p_i  <  i, c_i is lowercase English letter), that mean that there is an edge between nodes p_i and i and there is a letter c_i written on this edge.

Print n integers. The i-th of them should be the length of the longest simple path in subtree of the i-th vertex that form a Dokhtar-kosh string.

4
1 s
2 a
3 s

3 1 1 0 

5
1 a
2 z
1 a
4 z

4 1 0 1 0