We present a unification and generalization of what is known in the literature as sequentially and hierarchically semiseparable (SSS and HSS) representations for matrices. These so-called tree quasi-separable (TQS) matrices contain sparse matrices with tree-structured adjacency graphs as an important subcase. TQS matrices inherit all the favorable algebraic properties of SSS and HSS under addition, products, and inversion. To arrive at these properties, we prove a key result that characterizes the conversion of any dense matrix into a TQS representation. Here, we specifically show through an explicit construction that the size of the representation is dictated by the ranks of certain Hankel blocks of the matrix. Analogous to SSS and HSS, TQS matrices admit fast matrix-vector products and direct solvers. A sketch of the associated algorithms is provided. ... Read more