Algorithm

$1.$ Run DFS recording full tour: add $v$ when entering and when returning from each child.

$2.$ Store depth of each node in the tour.

$3.$ Record first[v] = first position of $v$ in the tour.

$4.$ Build RMQ structure on the depth array (sparse table for $O(1)$ query or segment tree for $O(\log n)$).

$5.$ For LCA($u$, $v$): query RMQ on range [min(first[u], first[v]), max(first[u], first[v])], return node at the position with minimum depth. Preprocessing: $O(n \log n)$. Query: $O(1)$ with sparse table or $O(\log n)$ with segment tree.

Space complexity is $O(n)$ for the data structures used.