Stacked directed animals and multi-directed animals defined without using heaps of pieces

Stacked directed animals and multi-directed animals are two lattice models defined by Bousquet-M\'{e}lou and Rechnitzer in 2002. The original definitions of those models involve heaps of pieces, i.e., some geometric representation of partially commutative monoids. The object of this writing is to define stacked directed animals and multi-directed animals in such a way that heaps of pieces are not involved. Our alternative definitions are equivalent to Bousquet-M\'{e}lou and Rechnitzer's original ones.