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

For any set $\mathrm{\Omega }$ of non-negative integers such that ${0,1}\subseteq \Omega$and${0,1}\ne \Omega$,weconsiderarandom$\Omega$-$k$-tree${\sf G}_{n,k}$thatisuniformlyselectedfromallconnected$k$-treesof$(n+k)$verticeswherethenumberof$(k+1)$-cliquesthatcontainanyfixed$k$-clique belongs to $\mathrm{\Omega }$. We prove that ${\mathsf{G}}_{n,k}$, scaled by $(k{H}_k{\sigma }_{\mathrm{\Omega }})\mathrm{/}(2\sqrt{n})$ where $H_k$ is the $k$-th Harmonic number and ${\sigma }_{\mathrm{\Omega }}>0$, converges to the Continuum Random Tree ${\mathcal{T}}_{\mathsf{e}}$. Furthermore, we prove the local convergence of the rooted random $\mathrm{\Omega }$-$k$-tree ${\mathsf{G}}_{n,k}^{\circ }$ to an infinite but locally finite random $\mathrm{\Omega }$-$k$-tree ${\mathsf{G}}_{\mathrm{\infty },k}$.