A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element ("datum") from an infinite set, where the latter can only be compared for equality. The article considers alternating automata on data trees that can move downward and rightward, and have one register for storing data. The main results are that nonemptiness over finite data trees is decidable but not primitive recursive, and that nonemptiness of safety automata is decidable but not elementary. The proofs use nondeterministic tree automata with faulty counters. Allowing upward moves, leftward moves, or two registers, each causes undecidability. As corollaries, decidability is obtained for two data-sensitive fragments of the XPath query language.