  We introduce a model of register automata over infinite trees with extrema
constraints. Such an automaton can store elements of a linearly ordered domain
in its registers, and can compare those values to the suprema and infima of
register values in subtrees. We show that the emptiness problem for these
automata is decidable.
  As an application, we prove decidability of the countable satisfiability
problem for two-variable logic in the presence of a tree order, a linear order,
and arbitrary atoms that are MSO definable from the tree order. As a
consequence, the satisfiability problem for two-variable logic with arbitrary
predicates, two of them interpreted by linear orders, is decidable.
