Suppose a tree with nodes located at levels $1,2,3...$. At each level the nodes branch into several nodes or do not branch.

Does the cardinality of the set of all infinite paths in this tree depend on the growth rate of the nodes by levels? Well, yes.

I mean, if the nodes do not branch at all, then the path is 1.

If the notes add a fixed amount of new nodes at each level, then the number of infinite paths seems to be countable. The same is if the number of nodes grows polynomially.

If the nodes branch at fixed rate at each level, e.g. each node gives 2 nodes, then the number of nodes grows exponentially and the set of all paths has cardinality of continuum.

What about intermediate branching rates?

Since there are intermediate growth rates between polynomial and exponential growth, can we make any conclusions about continuum hypothesis here? 
Will the most intermediate rates produce either continuum or countable cardinality of the paths anyway?