Logarithmic edge-length ratios for planar drawings of 2-trees

For a planar graph $G$ with at least one edge, let $\rho (G)$ be the infimum of the ratio of the longest to the shortest edge length over all planar straight-line drawings of $G$. We prove that every $n$-vertex $2$-tree $G$ satisfies $\rho (G)\le 4+5\lfloor {\log }_2(2n-3)\rfloor$. This improves the known $O(\sqrt{n})$ upper bound and, together with the lower bound of Blažej, Fiala and Liotta, shows that the maximum of $\rho (G)$ over $n$-vertex $2$-trees is $\mathrm{\Theta }(\log n)$. The proof adapts Biedl’s recursive bar construction using nonintegral heights and applies a height-preserving straightening theorem.