  We consider the problem of optimal approximation of a target measure by an
atomic measure with $N$ atoms, in branched optimal transport distance. This is
a new branched transport version of optimal quantization problems. New
difficulties arise, since in classical semi-discrete optimal transport with
Wasserstein distance, the interfaces between cells associated with neighboring
atoms have Vorono\"i structure and satisfy an explicit description. This
description is missing for our problem, in which the cell interfaces are
thought to have fractal boundary. We study the asymptotic behaviour of optimal
quantizers for absolutely continuous measures as the number $N$ of atoms grows
to infinity. We compute the limit distribution of the corresponding point
clouds and show in particular a branched transport version of Zador's theorem.
Moreover, we establish uniformity bounds of optimal quantizers in terms of
separation distance and covering radius of the atoms, when the measure is
$d$-Ahlfors regular. A crucial technical tool is the uniform in $N$ H\"older
regularity of the landscape function, a branched transport analog to
Kantorovich potentials in classical optimal transport.
