Inverse acyclic Z-matrices, matrices of tree structure and shifted generalized ultrametric matrices

Inverses of acyclic or treediagonal matrices are nicely characterized in the landmarked paper by Klein \cite{KLe82}. But also in \cite{Nab01} a characterization of these inverses is given which is related to the structure of inverses of tridiagonal matrices. Here we use this characterization to determine and construct matrices whose inverses are acyclic $Z$-matrices. Moreover, we establish relations between several classes of matrices, namely the classes of $Z$-matrices, inverse $Z$-matrices, ultrametric matrices, generalized ultrametric matrices, shifted ultrametric matrices, acyclic matrices, inverse acyclic matrices, matrices of tree structure, cyclopes and generalized cyclopes.