  We introduce global one-counter tree automata (GOCTA) which deviate from
usual counter tree automata by working on only one counter which is passed
through the tree in lexicographical order, rather than duplicating the counter
at every branching position. We compare the capabilities of GOCTA to those of
counter tree automata and obtain that their classes of recognizable tree
languages are incomparable. Moreover, we show that the emptiness problem of
GOCTA is undecidable while, in stark contrast, their membership problem is in
P.
