  The aim of this article is to study the forest composed by point-to-line
geodesics in the last-passage percolation model with exponential weights. We
will show that the location of the root can be described in terms of the maxima
of a random walk, whose distribution will depend on the geometry of the
substrate (line). For flat substrates, we will get power law behaviour of the
height function, study its scaling limit, and describe it in terms of
variational problems involving the Airy process.
