  We prove a new system of relations in the tautological ring of the moduli
space of curves involving stable rooted trees with level structure decorated by
the top Chern class of the Hodge bundle and $\Omega$-classes and double
ramification structures. In particular, this resolves a recent conjecture on
these relations as well as connects with one of the two sides of the recently
established DR/DZ equivalence between the integrable hierarchies constructions
of Buryak and of Dubrovin--Zhang.
