The Precedence-Constrained Minimum-Cost Arborescence problem, has been recently proposed. The purpose of the precedence constraints, that are enforced between pairs of vertices, is to prevent certain directed paths to appear in the tree that violate a precedence relationship. In this work we introduce a new mixed integer linear programming model that uses a smaller number of variables and constraints to model the precedence relationships compared to those previously appeared in the literature. Furthermore, two models with a polynomial number of variables and constraints are introduced. It is based on a network-flow formulation to model the connectivity of the arborescence. Extensive computational experiments have been run to validate the new models.