  In a previous paper Hua-Jost-Liu, we have applied Alexandrov geometry methods
to study infinite semiplanar graphs with nonnegative combinatorial curvature.
We proved the weak relative volume comparison and the Poincar\'e inequality on
these graphs to obtain an dimension estimate of polynomial growth harmonic
functions which is asymptotically quadratic in the growth rate. In the present
paper, instead of using volume comparison on graphs, we directly argue on
Alexandrov spaces to obtain the optimal dimension estimate of polynomial growth
harmonic functions on graphs which is actually linear in the growth rate. From
a harmonic function on the graph, we construct a function on the corresponing
Alexandrov surface that is not necessarily harmonic, but satisfies crucial
estimates.
