  Nominal set plays a central role in a group-theoretic extension of finite
automata to those over an infinite set of data values. Moerman et al. proposed
an active learning algorithm for nominal word automata with the equality
symmetry. In this paper, we introduce deterministic bottom-up nominal tree
automata (DBNTA), which operate on trees whose nodes are labelled with elements
of an orbit finite nominal set. We then prove a Myhill-Nerode theorem for the
class of languages recognized by DBNTA and propose an active learning algorithm
for DBNTA. The algorithm can deal with any data symmetry that admits least
support, not restricted to the equality symmetry and/or the total order
symmetry. To prove the termination of the algorithm, we define a partial order
on nominal sets and show that there is no infinite chain of orbit finite
nominal sets with respect to this partial order between any two orbit finite
sets.
