  We consider the slow movement of randomly biased random walk $(X_n)$ on a
supercritical Galton--Watson tree, and are interested in the sites on the tree
that are most visited by the biased random walk. Our main result implies
tightness of the distributions of the most visited sites under the annealed
measure. This is in contrast with the one-dimensional case, and provides, to
the best of our knowledge, the first non-trivial example of null recurrent
random walk whose most visited sites are not transient, a question originally
raised by Erd\H{o}s and R\'ev\'esz [11] for simple symmetric random walk on the
line.
