  We study a genealogical model for continuous-state branching processes with
immigration with a (sub)critical branching mechanism. This model allows the
immigrants to be on the same line of descent. The corresponding family tree is
an ordered rooted continuum random tree with a single infinite end defined
thanks to two continuous processes denoted by $(\overleftarrow{H}_t ;t\geq 0)$
and $(\overrightarrow{H}_t ;t\geq 0)$ that code the parts at resp. the left and
the right hand of the infinite line of descent of the tree. These processes are
called the left and the right height processes. We define their local time
processes via an approximation procedure and we prove that they enjoy a
Ray-Knight property. We also discuss the important special case corresponding
to the size-biased Galton-Watson tree in the continuous setting. In the last
part of the paper we give a convergence result under general assumptions for
rescaled discrete left and right contour processes of sequences of
Galton-Watson trees with immigration. We also provide a strong invariance
principle for a sequence of rescaled Galton-Watson processes with immigration
that also holds in the supercritical case.
