The acyclic matrices with a P-set of maximum size

Let mA(0) denote the nullity of a given matrix A of order n. Set A(α) for the principal submatrix of A obtained after deleting the rows and columns indexed by the nonempty subset α of {1,…,n}. When mA(α)(0)=mA(0)+|α|, we call α a P-set of A. In this paper, we classify all of the trees T for which there exists a matrix A whose graph is T and containing a P-set of maximum size. Our characterization does not depend on whether the acyclic matrices are singular or nonsingular.