The multivariate singularity analysis of Pemantle and Wilson is explored and then used to derive an asymptotic expression for the number of bicolored supertrees, realized as the diagonal sequence of a bivariate rational function F. These asymptotics have been obtained previously by univariate methods, but the analysis contained herein serves as a case study for the general multivariate method. The analysis itself relies heavily on the structure of a height function h along the pole set V of F.

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... hat makes this example interesting is the geometry of h on V, namely that h has a degenerate saddle point away from the boundary of the domain of analyticity of F that contributes to the asymptotic analysis. Due to the geometry of h at this point, the coefficient analysis can not be computed directly from the standard formulas of multivariate singularity analysis. Performing the analysis in this case represents a first step towards understanding general cases of this geometric type.