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

For any set Ω of non-negative integers such that $\{0,1\}⊊\Omega$ , we consider a random Ω-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 Ω. We prove that G_n,k, scaled by $\left(k{H}_k{\sigma }_{\Omega }\right)/\left(2\sqrt{n}\right)$ where H _k is the kth harmonic number and σ _Ω > 0, converges to the continuum random tree $T_e$ . Furthermore, we prove local convergence of the random Ω-k-tree $G_{n,k}^{\circ }$ to an infinite but locally finite random Ω-k-tree G_∞,k.