Height and contour processes of Crump-Mode-Jagers forests (III): The binary, homogeneous universality class

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.