The cardinality of the set of all root paths in the infinite complete binary tree is equal to the cardinality of the Continuum. The same holds true for k-ary trees for any finite k. But what is the case for k infinite?
You may be also interested in the following paper by Shelah: http://shelah.logic.at/files/589.pdf
In this paper (section 2) he defines the more general notion of the "tree revised power" of two cardinals k1, k2 as the supremum on the number of k2-branches of trees with k1 nodes. He then proves that certain inequalities involving the tree revised power have some interesting consequences in pcf theory.