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

For any set Ω of non-negative integers such that , we consider a random Ω-k-tree G 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 Ω. We prove that G, scaled by where H is the kth harmonic number and σ  > 0, converges to the continuum random tree . Furthermore, we prove local convergence of the random Ω-k-tree to an infinite but locally finite random Ω-k-tree G.