Polynomial graph invariants have been confirmed to have important
applications in quantum chemistry and biological information. One of the
famous polynomial graph invariants is the Tutte polynomial which gives
multifarious interesting information about the graph structure. In this
paper, we first give a simpler and more efficient method to get the
Tutte polynomials of alternating polycyclic chains. Then we obtain the
explicit expressions for the Tutte polynomials and the number of
spanning trees of phenylene systems with given number of branching
hexagons. Moreover, we determine the extremal values of the number of
spanning trees among the phenylene systems with given one or two
branching hexagons, and the corresponding extremal phenylene systems are
characterized, respectively.