We use profinite Bass-Serre theory (the theory of profinite group actions on profinite trees) to prove that the fundamental groups of finite graphs of free groups which are l-acylindrical and have finitely generated edge groups are conjugacy separable. We apply this theorem to: demonstrate that a generic positive one-relator group is conjugacy separable; produce a variant of the Rips con- struction in which the output group is conjugacy separable; apply this last to exhibit an example of a strong profinite equivalence between two finitely presented groups, one of which is conjugacy separable and the other having unsolvable conjugacy problem. We further use profinite Bass-Serre theory to demonstrate that having one end is an up-weak pro-C property for any extension- closed class C of finite groups. We show by example that it is not a down-weak pro-p property for any prime p. We consider Korenev's definition of pro-p ends for a pro-p group, and show that the number of ends of a finitely generated residually p group cannot be less than the number of pro-p ends of its pro-p completion. We explore possibilities for, but are ultimately unsuc- cessful in giving, a proper analogue of Stallings' theorem for pro-p groups. We ask which other properties might be profinite, and use another variant of the Rips construction to produce examples of patholog- ical groups such that either they are hyperbolic groups which are not residually finite, or neither property (FA) nor property (T) is an up-weak profinite property.