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Crump-Mode-Jagers (CMJ) trees generalize Galton-Watson trees by allowingindividuals to live for an arbitrary duration and give birth at arbitrary timesduring their life-time. In this paper, we are interested in the height andcontour processes encoding a general CMJ tree.  We show that the one-dimensional distribution of the height process can beexpressed in terms of a random transformation of the ladder height processassociated with the underlying Lukasiewicz path. As an application of thisresult, when edges of the tree are "short" we show that, asymptotically, (1)the height process is obtained by stretching by a constant factor the heightprocess of the associated genealogical Galton-Watson tree, (2) the contourprocess is obtained from the height process by a constant time change and (3)the CMJ trees converge in the sense of finite-dimensional distributions.

Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges