It turns out, that the final amortized time complexity is $$$O(α(n))$$$, where $$$α(n)$$$ is the inverse Ackermann function, which grows very slowly. In fact it grows so slowly, that it doesn't exceed $$$4$$$ for all reasonable $$$n$$$ (approximately $$$n<10^{600})$$$.

int find_set(int v) {
    if (v == parent[v])
        return v;
    return parent[v] = find_set(parent[v]);
}

Question: Why this works in $$$O(α(n))$$$?