Check Your Understanding

Quick complexity check: what is the time complexity of preprocessing the Euler tour?

Answer: $O(n)$ for the DFS traversal. What is the space complexity?

Answer: $O(n)$ for the $tin$ and $tout$ arrays, plus whatever data structure you layer on top. What about query time?

That depends on your data structure. Segment tree gives $O(\log n)$ per query. Fenwick tree also gives $O(\log n)$. The Euler tour itself adds no extra cost beyond the initial $O(n)$ preprocessing.