The notion of left convergence for graphs (in relation with homomorphism densities for fixed patterns and Szemer\'edi’s regularity lemma) introduced by Lov\' asz et al. got increasingly studied over the past $10$ years. Recently, Ne\v set\v ril and Ossona de Mendez introduced a general framework to study the limits of structures based on the converging probability for the structures of the sequence to verify any formula of a given fragment of first order logic for a random assignment of free variables. In this context, a sequence is {\em quantifier-free convergent} (or QF-convergent) if the probability of any equation being satisfied converges. We will give examples, and show how the QF-convergence of colored tree-semilattices can be related to the left convergence of $m$-partite cographs, a generalization of cographs.