Combinatorial representation in terms of Schroder paths and other weighted plane paths are given of Laurent biorthogonal polynomials ( LBPs) and a linear functional with which LBPs have orthogonality and biorthogonality. Particularly, it is clarified that quantities to which LBPs are mapped by the corresponding linear functional can be evaluated by enumerating certain kinds of Schroder paths, which imply orthogonality and biorthogonality of LBPs.