I suppose a definition for Infinite Tree would be required to answer this question. For my particular research topic, the following definition would be good for a tree:
Definition A tree is a graph for which there exist a unique path between any two vertices.
Note how this works for a vertex set of any cardinality (unless it doesn't, in which case I'm missing something and please feel free to point it out).
Now with this definition, does anyone know the answer to this question?