This is a boring, technical question that I stumbled upon while making a contribution to Sage. I would still like to hear a constructive answer so hopefully the question does not get closed. 

The question is the following.


  How many spanning trees does the empty graph $E$ have?


According to Sage it has 1, while Mathematica claims $\tau(E) = 0.$ Now the only subgraph of $E$ is $E$ hence this question can be rephrased as


  Is $E$ a tree?


One characterization says that a tree is a connected graph with $n$ vertices and $n-1$ edges and would imply that $E$ is not a tree. However if we define a tree as a connected acyclic graph then $E$ is clearly a tree.

It appears that as far as Kirchhoff is concerned any value would do since $$\rm{adj}(\mathcal{L}(E)) = \mathcal{L}(E) = k\mathcal{L}(E)$$ for any $k.$

Hence what I am wondering is


  Are there any wider reasons in defining $E$ to (not) be a tree?


In a paper "Is the null-graph a pointless concept?" Harary and Read examine reasons for
assigning certain properties to the empty graph. They observe that from the enumeration perspective it appears to be convenient to consider the empty graph as a forest, but not a tree.