We have a graph G of treewidth $\operatorname{tw}(G)\leq k$, for some $k\in\mathbb{N}$. I've seen a claim that that implies that the clique-width of the same graph is at most $k \cdot 2^k$. This implies that given a tree decomposition of the graph we can construct a proper expression tree using at most $k \cdot 2^k$ labels.
I'm guessing that the $k$ in the decomposition allows us to "construct" a node of the tree decomposition, i.e. regardless how the vertices in one bag are connected, we can easily connect them since we have $k+1$-ish labels at our disposal. However, I don't quite see how we can use that fact to construct the complete expression tree (and therefore the $k\cdot 2^k$-expression).
How can we construct the proper expression and therefore prove that graphs of bounded treewidth are a subclass of graphs of bounded clique-width
