The Find Operation

To find the leader of node $u$:

$1.$ Check if parent[u] == u.

If yes, $u$ is the leader. Return $u$.

$2.$ If no, move up: $u$ = parent[u], and repeat.

You climb the tree until you reach a root. The root is the leader of $u$'s group. This takes $O(h)$ time where $h$ is tree height. Without optimizations, $h$ could be $O(n)$. With path compression, $h$ stays tiny.

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