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

For any set Ω of non‐negative integers such that { 0 , 1 } ⊊ Ω , 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 ( k H k σ Ω ) / ( 2 n ) where H k is the k th harmonic number and σ Ω  > 0, converges to the continuum random tree T e . Furthermore, we prove local convergence of the random Ω‐ k ‐treeG n , k ∘to an infinite but locally finite random Ω‐ k ‐tree G ∞, k .