  We are interested in the structure of large Bienaym\'e-Galton-Watson random
trees whose offspring distribution is critical and falls within the domain of
attraction of a stable law of index $\alpha=1$. In stark contrast to the case
$\alpha \in (1,2]$, we show that a condensation phenomenon occurs: in such
trees, one vertex with macroscopic degree emerges. To this end, we establish
limit theorems for centered downwards skip-free random walks whose steps are in
the domain of attraction of a Cauchy distribution, when conditioned on a late
entrance in the negative real line. These results are of independent interest.
As an application, we study the geometry of the boundary of random planar maps
in a specific regime (called non-generic of parameter $3/2$). This supports the
conjecture that faces in Le Gall & Miermont's $3/2$-stable maps are
self-avoiding.
