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

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