What The Alpha Needs

With both optimizations the amortized cost is O(alpha(n)), and alpha(n) stays at or below 4 for every n a real program will see (roughly up to $10^{600}$).

Path compression alone already reaches the O(alpha(n)) amortized bound; union by rank only lowers the constant factor.

Union by size or by rank, with no path compression at all, still gives O(log n) per query.

Path compression on its own, with unions that always hang the second root under the first, gives O(log n) per call on average.

With both optimizations, every individual find_set call runs in O(alpha(n)) time.