Certain Residual Properties of Bounded Nilpotent Groups and Their Tree Products

Let $$\mathfrak{P}$$ be a nonempty set of primes. We prove that any $$\mathfrak{P}$$ -bounded nilpotent group is $$\mathfrak{P}$$ -potent and the tree product T of a finite number of $$\mathfrak{P}$$ -bounded nilpotent groups with proper locally cyclic edge subgroups is residually a finite $$\mathfrak{P}$$ -group iff any vertex group of T has no $$\mathfrak{P}{\kern 1pt} '$$ -torsion and any edge subgroup of T is $$\mathfrak{P}{\kern 1pt} '$$ -isolated in the vertex group containing it. We prove also that the tree product of a finite number of groups with locally cyclic edge subgroups is residually a finite $$p$$ -group if all its vertex groups have this property and any edge subgroup is separable in the corresponding vertex group by the class of finite $$p$$ -groups.