2791 - Count Paths That Can Form a Palindrome in a Tree\.

Hard

You are given a tree (i.e. a connected, undirected graph that has no cycles) rooted at node 0 consisting of n nodes numbered from 0 to n - 1. The tree is represented by a 0-indexed array parent of size n, where parent[i] is the parent of node i. Since node 0 is the root, parent[0] == -1.

You are also given a string s of length n, where s[i] is the character assigned to the edge between i and parent[i]. s[0] can be ignored.

Return the number of pairs of nodes (u, v) such that u < v and the characters assigned to edges on the path from u to v can be rearranged to form a palindrome.

A string is a palindrome when it reads the same backwards as forwards.

Example 1:

Input: parent = -1,0,0,1,1,2, s = "acaabc"

Output: 8

Explanation:

The valid pairs are:

• All the pairs (0,1), (0,2), (1,3), (1,4) and (2,5) result in one character which is always a palindrome.

• The pair (2,3) result in the string "aca" which is a palindrome.

• The pair (1,5) result in the string "cac" which is a palindrome.

• The pair (3,5) result in the string "acac" which can be rearranged into the palindrome "acca".

Example 2:

Input: parent = -1,0,0,0,0, s = "aaaaa"

Output: 10

Explanation: Any pair of nodes (u,v) where u < v is valid.

Constraints:

• n == parent.length == s.length

• <code>1 <= n <= 10⁵</code>

• 0 <= parent[i] <= n - 1 for all i >= 1

• parent[0] == -1

• parent represents a valid tree.

• s consists of only lowercase English letters.