  We build an example of a system $\mathcal{S}$ of similarities in
$\mathbb{R}^2$ whose attractor is a plane dendrite $K\supset [0,1]$ which
satisfies one point intersection property, while the post-critical set of the
system $\mathcal{S}$ is a countable set whose natural projection to $K$ is
dense in the middle-third Cantor set.
