Perhaps one might hope to aim for a tree whose set of paths has cardinality between $\omega$ and $2^\omega$, when the CH fails, for in this case there would be many such intermediate cardinalities at which to take aim. But alas, this will be impossible. This is because the set of paths through such a tree is a closed set, and Cantor proved that every closed set is either countable or has cardinality continuum, and never between. This can be seen via the [Cantor-Bendixson theorem](http://en.wikipedia.org/wiki/Perfect_set_property), which shows that every closed set is the union of a countable set (obtained by transfinitely throwing out isolated points in the Cantor-Bendixson derivatives) and a perfect set, which if nonempty, has size continuum. 

Thus, one cannot hope to get cardinalities between $\omega$ and $2^\omega$ this way, even if there are many cardinalities in this interval.