Disjoint-set (DSU) / Union-Find

namespace DSU {
    /**
    * @brief: vector `p` contains -(size of tree) if p is a  root else ID of parent in tree.
    */
    class DSU {
    public:
        vector<int> p;

        DSU(int32_t n) { p.assign(n, -1); }

        int root(int32_t v) { return (p[v] < 0) ? v : p[v] = root(p[v]); }
        bool same(int32_t u, int32_t v) { return root(u) == root(v); }

        /**
        * @return: `true` if u, v are not already in same tree.
        */
        inline bool join(int32_t u, int32_t v) {
            if ((u = root(u)) == (v = root(v))) return false;
            if (p[u] > p[v]) swap(u, v);
            p[u] += p[v]; p[v] = u; return true;
        }
    };
}

Notes

The root node of the Tree contains the -(size of subtree) . All other nodes contain Parent ID.
The DSU::root function can also be implemented iteratively as below

int root(int32_t v) {
  if (p[v] < 0) return v;
  int r = v; while (p[r] >= 0) r = p[r];
  while (p[v] >= 0) { auto pa = p[v]; p[v] = r; v = pa; }
  return r;
}

Notes​

Notes