# Clique-width expressions with logarithmic depth

When we are given a tree decomposition of a graph $G$ with width $w$, there are several ways in which we can make it "nice". In particular, it is known that it is possible to transform it into a tree decomposition where the tree is binary and its height is $O(\log n)$. This can be achieved while keeping the width of the decomposition at most $3w$. (See e.g. "Parallel algorithms with optimal speedup for bounded treewidth", by Bodlaender and Hagerup). So, logarithmic depth is a property of a tree decomposition which we can get almost for free.

My question is if there exists a similar result for clique-width, or perhaps a counter-example. In other words, given a clique-width expression for $G$ using $k$ labels, does there always exist a clique-width expression of height $O(\log n)$ for $G$, that uses at most $f(k)$ labels? Here, the height is defined naturally as the height of the parse tree of the clique-width expression.

If a statement similar to the above is not known, is there an example of an $n$-vertex graph $G$ with small clique-width $k$, such that the only way to construct $G$ with $f(k)$ labels is to use an expression with large depth?

"Every graph of clique-width or NLC-width $k$ is the value of a 3-balanced clique-width expression of clique-width or NLC-width at most $k \times 2^{k+1}$"