A Self-Similar Dendrite with One-Point Intersection and Infinite Post-Critical Set

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.