Assuming your path has countable length, the set of all paths in a $k$-ary tree will have cardinality $k^{\aleph_0}=\max(k,2^{\aleph_0})$. 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]$). 1 Assuming your path has countable length, the set of all paths in a$k$-ary tree will have cardinality$k^{\aleph_0}=\max(k,2^{\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]\$).