Le Roux and Ziegler asked whether every simply connected compact nonempty planar
$\Pi^0_1$
set always contains a computable point. In this paper, we solve the problem of le Roux and Ziegler by showing that there exists a planar
$\Pi^0_1$
dendroid without computable points. We also provide several pathological examples of tree-like
$\Pi^0_1$
continua fulfilling certain global incomputability properties: there is a computable dendrite which does not *-include a
$\Pi^0_1$
tree; there is a
$\Pi^0_1$
dendrite which does not *-include a computable dendrite; there is a computable dendroid which does not *-include a
$\Pi^0_1$
dendrite. Here, a continuum A *-includes a member of a class
$\mathcal{P}$
of continua if, for every positive real ε, A includes a continuum
$B \in \mathcal{P}$
such that the Hausdorff distance between A and B is smaller than ε.
