  Nesetril and Ossona de Mendez introduced the notion of first order
convergence as an attempt to unify the notions of convergence for sparse and
dense graphs. It is known that there exist first order convergent sequences of
graphs with no limit modeling (an analytic representation of the limit). On the
positive side, every first order convergent sequence of trees or graphs with no
long path (graphs with bounded tree-depth) has a limit modeling. We strengthen
these results by showing that every first order convergent sequence of plane
trees (trees with embeddings in the plane) and every first order convergent
sequence of graphs with bounded path-width has a limit modeling.
