  We revisit the approximation of nonempty compact planar sets by filled-in
Julia sets of polynomials developed by Lindsey and Younsi and analyze the rate
of approximation. We use slightly modified fundamental Lagrange interpolation
polynomials and show that taking certain classes of nodes with subexponential
growth of Lebesgue constants improves the approximation rate. To this end we
investigate properties of some arrays of points in $\mathbb{C}$. In particular
we prove subexponential growth of Lebesgue constants for pseudo Leja sequences
with bounded Edrei growth on finite unions of quasiconformal arcs. Finally, for
some classes of sets we estimate more precisely the rate of approximation by
filled-in Julia sets in Hausdorff and Klimek metrics.
