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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 $[k]$).