A10749. 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.

The first line contains integer $n$ ( $1<=n<=5·10^{5}$ ) — the number of vertices in the tree.

$(n-1)$ lines follow, the $i$ -th of them contain an integer $p_{i+1}$ and a letter $c_{i+1}$ ( $1<=p_{i+1}<=i$ , $c_{i+1}$ is lowercase English letter, between a and v, inclusively), that mean that there is an edge between nodes $p_{i+1}$ and $i+1$ and there is a letter $c_{i+1}$ 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.