<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 UF:
    def __init__(self, N) -> None:
        self.p, self.size = list(range(N)), [1] * N

    def find(self, x):
        if self.p[x] != x: self.p[x] = self.find(self.p[x])
        return self.p[x]

    def connected(self, x, y): return self.find(x) == self.find(y)
    
    def connect(self, x, y):
        px = self.find(self.p[x])
        py = self.find(self.p[y])

        if px != py:
            if self.size[px] >= self.size[py]:
                self.p[py] = self.p[px]
                self.size[px] += self.size[py]
            else:
                self.p[px] = self.p[py]
                self.size[py] += self.size[px]

    def get_size(self, x): return self.size[self.find(x)]

class Solution:
    def get_primes(self, n):
        primes = [True] * (n+1)
        primes[0] = primes[1] = False
        for i in range(2, n+1):
            if primes[i]:
                for j in range(i*i, n+1, i): primes[j] = False
        return primes

    def countPaths(self, n, edges) -> int:
        g = defaultdict(set)
        prime = self.get_primes(n)
        uf = UF(n + 1)
        for u, v in edges:
            g[u].add(v)
            g[v].add(u)
            if prime[u] or prime[v]: continue
            uf.connect(u, v)
   
        def count_nodes(node):
            count = 0
            for nei in g[node]:
                if not prime[nei]: count += uf.get_size(nei)
            return count

        result = 0

        for i in range(1, n + 1):
            if prime[i]:
                count_of_nodes = count_nodes(i)

                for nei in g[i]:
                    if prime[nei]: continue
                    result += uf.get_size(nei)
                    count_of_nodes -= uf.get_size(nei)
                    result += count_of_nodes * uf.get_size(nei) 

        return result
1571