最近有幸作为面试观察员（面试shadow），参与了一场谷歌的软件工程师面试。面试中，候选人遇到了一道非常有趣的编程题目，难度相当高，涉及到了图论和数学知识。这题同时也是LeetCode 2867 题。

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 edges[i] = [ui, vi] indicates that there is an edge between nodes ui and vi 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.

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.

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:

• 1 <= n <= 105
• edges.length == n - 1
• edges[i].length == 2
• 1 <= ui, vi <= n
• The input is generated such that edges represent a valid tree.

这道题目考查的是候选人对树的遍历算法、质数判断以及图论中路径问题的理解和应用能力。我注意到，面试官在提出问题后，特别强调了对算法效率的考量，希望候选人能提出时间复杂度和空间复杂度都较优的解决方案。

候选人首先花了一些时间理解题目，确认了几个关键点后，开始思考解题思路。他首先分析了如何在树中找到所有可能的路径，接着探讨了如何高效判断一个数是否为质数。在确认解题思路后，候选人开始编写代码，过程中与面试官进行了几次交流，讨论了算法的有效性和可能的优化方案。

在我们的面试辅助中，我们专注于帮助求职者准备北美公司的在线评估和面试过程。我们的服务包括深入的面试技巧训练、简历和求职信的优化建议，以及定制的职业规划建议，面试代面，面试即时辅助，旨在帮助求职者在这些具有吸引力的北美科技巨头公司中脱颖而出。

无论您是寻求在IT领域发展的经验丰富的专业人士，还是刚刚开始职业生涯的应届毕业生，我们的服务都能提供必要的支持和指导，帮助您实现职业目标。联系我，让我们一起开启您在北美的成功求职之旅。