  This paper belongs to a series of papers aiming to investigate scaling limits
of Crump-Mode-Jagers (CMJ) trees. In the previous two papers we identified
general conditions under which CMJ trees belong to the universality class of
Galton-Watson and Bellman-Harris processes. In this paper we identify general
conditions for CMJ trees to belong to the universality class of binary,
homogeneous CMJ trees. These conditions state that the offspring process should
'look like' a renewal process, and also that it should not accumulate too many
atoms near the origin. We show in particular that any renewal process satisfies
these conditions.
