  We focus on counting the number of labeled graphs on $n$ vertices and
treewidth at most $k$ (or equivalently, the number of labeled partial
$k$-trees), which we denote by $T_{n,k}$. So far, only the particular cases
$T_{n,1}$ and $T_{n,2}$ had been studied. We show that $$ \left(c \cdot
\frac{k\cdot 2^k \cdot n}{\log k} \right)^n \cdot 2^{-\frac{k(k+3)}{2}} \cdot
k^{-2k-2}\ \leq\ T_{n,k}\ \leq\ \left(k \cdot 2^k \cdot n\right)^n \cdot
2^{-\frac{k(k+1)}{2}} \cdot k^{-k}, $$ for $k > 1$ and some explicit absolute
constant $c > 0$. The upper bound is an immediate consequence of the well-known
number of labeled $k$-trees, while the lower bound is obtained from an explicit
algorithmic construction. It follows from this construction that both bounds
also apply to graphs of pathwidth and proper-pathwidth at most $k$.
