Splitting theorems for pro-p groups acting on pro-p trees

Let G be an infinite finitely generated pro-p group acting on a pro-p tree such that the restriction of the action to some open subgroup is free. We prove that G splits over an edge stabilizer either as an amalgamated free pro-p product or as a pro-p ({text {HNN}})-extension. Using this result, we prove under a certain condition that free pro-p products with procyclic amalgamation inherit from its amalgamated free factors the property of each 2-generated pro-p subgroup being free pro-p. This generalizes known pro-p results, as well as some pro-p analogues of classical results in abstract combinatorial group theory.