  We present a unification and generalization of sequentially and
hierarchically semi-separable (SSS and HSS) matrices called tree semi-separable
(TSS) matrices. Our main result is to show that any dense matrix can be
expressed in a TSS format. Here, the dimensions of the generators are specified
by the ranks of the Hankel blocks of the matrix. TSS matrices satisfy a
graph-induced rank structure (GIRS) property. It is shown that TSS matrices
generalize the algebraic properties of SSS and HSS matrices under addition,
products, and inversion. Subsequently, TSS matrices admit linear time
matrix-vector multiply, matrix-matrix multiply, matrix-matrix addition,
inversion, and solvers.
