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

<p>Return <em>the <strong>number of valid paths</strong> in the tree</em>.</p>

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

<p><strong>Note</strong> that:</p>

<ul>
	<li>The path <code>(a, b)</code> is a sequence of <strong>distinct</strong> nodes starting with node <code>a</code> and ending with node <code>b</code> such that every two adjacent nodes in the sequence share an edge in the tree.</li>
	<li>Path <code>(a, b)</code> and path <code>(b, a)</code> are considered the <strong>same</strong> and counted only <strong>once</strong>.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/08/27/example1.png" style="width: 440px; height: 357px;" />
<pre>
<strong>Input:</strong> n = 5, edges = [[1,2],[1,3],[2,4],[2,5]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> 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.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/08/27/example2.png" style="width: 488px; height: 384px;" />
<pre>
<strong>Input:</strong> n = 6, edges = [[1,2],[1,3],[2,4],[3,5],[3,6]]
<strong>Output:</strong> 6
<strong>Explanation:</strong> 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.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>edges.length == n - 1</code></li>
	<li><code>edges[i].length == 2</code></li>
	<li><code>1 &lt;= u<sub>i</sub>, v<sub>i</sub> &lt;= n</code></li>
	<li>The input is generated such that <code>edges</code> represent a valid tree.</li>
</ul>

class Solution:
    def countPaths(self, n: int, edges: List[List[int]]) -> int:
        graph = [[] for _ in range(n + 1)]
        for u, v in edges:
            graph[u].append(v)
            graph[v].append(u)

        def isPrime(x):
            if x <= 1:
                return False
            elif x == 2:
                return True
            else:
                for i in range(2, int(sqrt(x))+1):
                    if (x % i == 0):
                        return False
            return True
        prime = [isPrime(i) for i in range(n + 1)]

        def solve(u, parent):
            path_with_prime, path_no_prime, ans = 0, 0, 0
            pwp_array = []
            pwp_sum = 0
            pnp_array = []
            pnp_sum = 0
            for v in graph[u]:
                if v != parent:
                    pwp, pnp, subAns = solve(v, u)
                    pwp_array.append(pwp)
                    pwp_sum += pwp
                    pnp_array.append(pnp)
                    pnp_sum += pnp
                    ans += subAns
            if prime[u]:
                prefix = 0
                for x in pnp_array:
                    ans += x * prefix + x
                    prefix += x
                ans += 1
            else:
                for j, x in enumerate(pwp_array):
                    ans += x * (pnp_sum - pnp_array[j]) + x
            if prime[u]:
                path_with_prime = 1 + pnp_sum
                path_no_prime = 0
            else:
                path_with_prime = pwp_sum
                path_no_prime = 1 + pnp_sum
            #print(path_with_prime, path_no_prime, ans, u)
            return path_with_prime, path_no_prime, ans
        return solve(1, -1)[2] - sum(prime)
            
1688