  We analyze a class of spatial random spanning trees built on a realization of
a homogeneous Poisson point process of the plane. This tree has a simple radial
structure with the origin as its root. We first use stochastic geometry
arguments to analyze local functionals of the random tree such as the
distribution of the length of the edges or the mean degree of the vertices. Far
away from the origin, these local properties are shown to be close to those of
a variant of the directed spanning tree introduced by Bhatt and Roy. We then
use the theory of continuous state space Markov chains to analyze some nonlocal
properties of the tree, such as the shape and structure of its semi-infinite
paths or the shape of the set of its vertices less than $k$ generations away
from the origin. This class of spanning trees has applications in many fields
and, in particular, in communications.
