Implementation - Find with Compression

You use recursion to implement Path Compression neatly.

function find(i):
    if parent[i] == i:
        return i
    parent[i] = find(parent[i])
    return parent[i]

The part parent[i] = find(...) is what flattens the tree.

This runs in $O(\alpha(n))$ amortized time and uses $O(n)$ space.