Graph limits of random graphs from a subset of connected k-trees

For any set of non-negative integers such that {0,1}, we consider a random -k-tree G(n,k) that is uniformly selected from all connected k-trees of (n + k) vertices such that the number of (k + 1)-cliques that contain any fixed k-clique belongs to . We prove that G(n,k), scaled by (kHk sigma)/(2n) where H-k is the kth harmonic number and sigma > 0, converges to the continuum random tree Te. Furthermore, we prove local convergence of the random -k-tree Gn,k circle to an infinite but locally finite random -k-tree G(infinity,k).