We prove that, among rectangular grid graphs with a fixed number of vertices, the number of spanning trees increases when the side lengths are made more balanced. In particular, among all rectangular grid graphs with $n^2$ vertices, the square $n\times n$ grid has the largest number of spanning trees. The proof starts with the Laplacian product formula, passes to hyperbolic coordinates, and compares logarithms by separating a discrete-concavity term from a positive decreasing residual term.

翻译：我们证明：在顶点数固定的矩形网格图中，当边长变得更均衡时，生成树的数量会增加。特别地，在所有包含$n^2$个顶点的矩形网格图中，$n\times n$正方形网格具有最大数量的生成树。证明从拉普拉斯乘积公式出发，通过双曲坐标变换，并通过从正递减残差项中分离出离散凹性项来比较对数。