We observe that pushdown tree automata (PTAs) known in the literature cannot express combinations of branching and pushdown properties. This is because a PTA processes the children of a tree node in possibly different control states but with identical stacks. We propose branching pushdown tree automata (BPTAs) as a solution. In a BPTA, a push-move views its matching pops as an unbounded, unordered set of successor moves and can assert existential and universal requirements on them, just the way finite automata on unranked, unordered trees pass requirements to the children of a tree node. We show that BPTAs can express some natural properties and are more expressive than PTAs. Using a small-model theorem, we prove their emptiness problem to be decidable. The problem becomes undecidable, however, if push-moves are allowed to specify the ordering of matching pops.