Subtrees as Ranges

Here is what matters: node $u$ is in $v$'s subtree if and only if tin[v] <= tin[u] <= tout[v]. Why?

DFS enters $v$ before any descendant and exits $v$ after all descendants. The interval [tin[v], tout[v]] contains all entry times of nodes in $v$'s subtree. If you store node values at position tin[v] in an array, every subtree becomes a contiguous range. Subtree queries become range queries.