We study a hierarchy of variants of K\"onig's lemma for infinite
binary trees with finitely many infinite paths, in the context of the
Weihrauch degrees. The principle that every Cauchy real has a binary
expansion is shown to be constructively equivalent to K\"onig's lemma
for infinite binary trees in which every level has at most two nodes,
which is also Weihrauch equivalent to K\"onig's lemma for infinite
binary trees with at most two paths.