2867 - Count Valid Paths in a Tree\.

Hard

There is an undirected tree with n nodes labeled from 1 to n. You are given the integer n and a 2D integer array edges of length n - 1, where <code>edgesi = u_i, v_i</code> indicates that there is an edge between nodes <code>u_i</code> and <code>v_i</code> in the tree.

Return the number of valid paths in the tree.

A path (a, b) is valid if there exists exactly one prime number among the node labels in the path from a to b.

Note that:

• The path (a, b) is a sequence of distinct nodes starting with node a and ending with node b such that every two adjacent nodes in the sequence share an edge in the tree.

• Path (a, b) and path (b, a) are considered the same and counted only once.

Example 1:

Input: n = 5, edges = \[\[1,2],1,3,2,4,2,5]

Output: 4

Explanation: The pairs with exactly one prime number on the path between them are:

• (1, 2) since the path from 1 to 2 contains prime number 2.

• (1, 3) since the path from 1 to 3 contains prime number 3.

• (1, 4) since the path from 1 to 4 contains prime number 2.

• (2, 4) since the path from 2 to 4 contains prime number 2.

It can be shown that there are only 4 valid paths.

Example 2:

Input: n = 6, edges = \[\[1,2],1,3,2,4,3,5,3,6]

Output: 6

Explanation: The pairs with exactly one prime number on the path between them are:

• (1, 2) since the path from 1 to 2 contains prime number 2.

• (1, 3) since the path from 1 to 3 contains prime number 3.

• (1, 4) since the path from 1 to 4 contains prime number 2.

• (1, 6) since the path from 1 to 6 contains prime number 3.

• (2, 4) since the path from 2 to 4 contains prime number 2.

• (3, 6) since the path from 3 to 6 contains prime number 3.

It can be shown that there are only 6 valid paths.

Constraints:

• <code>1 <= n <= 10⁵</code>

• edges.length == n - 1

• edges[i].length == 2

• <code>1 <= u_i, v_i<= n</code>

• The input is generated such that edges represent a valid tree.