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

For any set Ω of non‐negative integers such that [Formula: see text] , 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 [Formula: see text] where H (k) is the kth harmonic number and σ (Ω) > 0, converges to the continuum random tree [Formula: see text]. Furthermore, we prove local convergence of the random Ω‐k‐tree [Formula: see text] to an infinite but locally finite random Ω‐k‐tree G(∞,k).