  For any set $\Omega$ of non-negative integers such that $\{0,1\}\subseteq
\Omega$ and $\{0,1\}\ne \Omega$, we consider a random $\Omega$-$k$-tree ${\sf
G}_{n,k}$ that is uniformly selected from all connected $k$-trees of $(n+k)$
vertices where the number of $(k+1)$-cliques that contain any fixed $k$-clique
belongs to $\Omega$. We prove that ${\sf G}_{n,k}$, scaled by
$(kH_{k}\sigma_{\Omega})/(2\sqrt{n})$ where $H_{k}$ is the $k$-th Harmonic
number and $\sigma_{\Omega}>0$, converges to the Continuum Random Tree
$\mathcal{T}_{{\sf e}}$. Furthermore, we prove the local convergence of the
rooted random $\Omega$-$k$-tree ${\sf G}_{n,k}^{\circ}$ to an infinite but
locally finite random $\Omega$-$k$-tree ${\sf G}_{\infty,k}$.
