In this paper we consider the properties of linear systems by means of directed graphs and numerical structures. We also state efficient algorithms for determining an approximate number of the non-zero terms within determinants' expressions of their matrices. The stated algorithms make use of trees representing numerical structures which contains the indices of the nonzero terms. This paper yields interesting results used in practical engineering applications which include linear systems with sparse matrices, for example: networks, electronic circuits, earth velocities boxes (gearboxes), multi-works systems ...etc.