The first diameter of $3$-manifolds of positive scalar curvature

We seek a universal upper bound for the first diameter of $3$-manifolds of scalar curvature $\geq + 1$. We find it in the case of finite fundamental group by using a combinatorial theorem about finite trees, and in the case when ${\pi _1}$ is infinite cyclic by using a weak notion of a ${\pi _1}$-equivariant Busemann function.