A connected graph has many spanning trees. Choose one uniformly at random. How likely is it to be isomorphic to any particular tree T?
Bitonti, Michel, and Scott (arXiv:2603.17630) prove that in a connected graph with n vertices and minimum degree d, no single tree shape dominates. The probability that a uniformly random spanning tree is isomorphic to any fixed tree T is at most n^{-Ω(d)}. Equivalently, the graph contains at least n^{Ω(d)} non-isomorphic spanning trees.
The bound is tight up to constant factors, confirming a conjecture by Lee in its strongest form.
The structural insight is that minimum degree forces diversity. A graph where every vertex has many neighbors cannot funnel its spanning trees into a small number of shapes. The combinatorial paths through the graph are too varied — each vertex's high degree provides choices that compound multiplicatively across the tree, producing exponentially many distinct topologies.
This is an anticoncentration result. Concentration inequalities bound how far a random variable strays from its mean. Anticoncentration bounds say the opposite: the random variable cannot concentrate too heavily on any single value. Here the “value” is a tree isomorphism class, and the “random variable” is the shape of a uniformly random spanning tree.
The result quantifies an intuition: well-connected graphs are structurally rich. Not just in the number of spanning trees (Kirchhoff's theorem counts those), but in the number of structurally distinct spanning trees. Counting is not the same as classifying. A graph could have exponentially many spanning trees that are all isomorphic to each other. High minimum degree prevents this collapse of diversity.