  The notion of left convergent sequences of graphs introduced by Lov\' asz et
al. (in relation with homomorphism densities for fixed patterns and
Szemer\'edi's regularity lemma) got increasingly studied over the past $10$
years. Recently, Ne\v set\v ril and Ossona de Mendez introduced a general
framework for convergence of sequences of structures. In particular, the
authors introduced the notion of $QF$-convergence, which is a natural
generalization of left-convergence. In this paper, we initiate study of
$QF$-convergence for structures with functional symbols by focusing on the
particular case of tree semi-lattices. We fully characterize the limit objects
and give an application to the study of left convergence of $m$-partite
cographs, a generalization of cographs.
