Is a "tree" with $0$ vertices, $0$ edges or $1$ vertex, $0$ edges considered a valid tree?
Here are a few properties of trees which can be relevant:


A tree on $n$ vertices has $n-1$ edges. Works for (2) but not for (1).
A graph has no cycles iff it is a disjoint union of trees. Works for both (1) and (2), unless you require the decomposition into trees to be unique, in which case it works for (2) but not for (1).
A tree is either a single vertex or a bunch of disjoint trees connected to a common vertex. Works for (2) but not for (1).


These examples suggest that the commonsense definition of tree should include (2) but not (1). Indeed, oftentimes it seems that the graph on no vertices is a pointless concept; see the classical discussion of Harary and Read, who present arguments for both sides of the issue.

Practically speaking, you should probably assume that (1) is not a legal graph, unless it makes sense in your application, and then you should explicitly mention the fact.