  We prove a general Ramsey theorem for trees with a successor operation. This
theorem is a common generalization of the Carlson-Simpson Theorem and the
Milliken Tree Theorem for regularly branching trees.
  Our theorem has a number of applications both in finite and infinite
combinatorics. For example, we give a short proof of the unrestricted
Ne\v{s}et\v{r}il-R\"odl theorem, and we recover the Graham-Rothschild theorem.
Our original motivation came from the study of big Ramsey degrees - various
trees used in the study can be viewed as trees with a successor operation. To
illustrate this, we give a non-forcing proof of a theorem of Zucker on big
Ramsey degrees.
