We recall the classical tree-cotree technique in magnetostatics. (1) We extend it in the frame of high-order finite elements in general domains. (2) We focus on its connection with the question of the invertibility of the final algebraic system arising from a high-order edge finite element discretization of the magnetostatic problem formulated in terms of the magnetic vector potential. With the same purpose of invertibility, we analyse another classically used condition, the Coulomb gauge. (3) We conclude by underlying that the two gauges can be naturally considered in a high order framework without any restriction on the topology of the domain.