The number of spanning trees $\tau(G)$ of a simple graph $G$ is seen to satisfy the deletion-contraction recurrence: $$\tau(G)=\tau(G-e)+\tau(G.e),$$ where $e$ is an edge of the graph $G$ and $G-e$ denotes the graph $G$ with edge $e$ deleted and $G.e$ denotes the graph $G$ with the edge $e$ contracted.
Now, the chromatic polynomial $P(G)$ also satisfies a similar deletion-contraction relation: $$P(G)=P(G-e)-P(G.e).$$ This gives rise to the question that is there some sort of a relation between the chromatic properties and the number of spanning trees in a graph $G$. Specifically,
Is there a relation between the number of spanning trees and the chromatic number of a graph?
Are they at least asymptotically related?