  Erd\H{o}s and R\'ev\'esz initiated the study of favorite sites by considering
the one-dimensional simple random walk. We investigate in this paper the same
problem for a class of null-recurrent randomly biased walks on a supercritical
Gaton-Watson tree. We prove that there is some parameter $\kappa \in (1,
\infty]$ such that the set of the favorite sites of the biased walk is almost
surely bounded in the case $\kappa \in (2, \infty]$, tight in the case
$\kappa=2$, and oscillates between a neighborhood of the root and the boundary
of the range in the case $\kappa \in (1, 2)$. Moreover, our results yield a
complete answer to the cardinality of the set of favorite sites in the case
$\kappa \in (2, \infty]$. The proof relies on the exploration of the Markov
property of the local times process with respect to the space variable and on a
precise tail estimate on the maximum of local times, using a change of measure
for multi-type Galton-Watson trees.
