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? 



Cardinality of the set of all paths in the infinite complete infinitary tree
Assuming your path has countable length, the set of all paths in a $k$-ary tree will have cardinality $k^{\aleph\_0}$. Indeed, at each step you have $k$ choices, and there are $\aleph\_0$ steps (think of a path as a function from $\mathbb{N}$ to $[k]$).


