We study the growth rate of the number of spanning trees of a sequence of planar graph which diagrammatically converge to a biperiodic planar graph. We relate this growth rate to the Mahler measure of 2-variable polynomials and hyperbolic volume of link complements. We use this circle of ideas to study interesting conjectures in knot theory and graph theory which relate hyperbolic volume and asymptotic growth rate of spanning trees.