  We determine the tropicalizations of very affine surfaces over a valued field
that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing
their (-1)-curves. On these tropical surfaces, the boundary divisors are
represented by trees at infinity. These trees are glued together according to
the Petersen, Clebsch and Schl\"afli graphs, respectively. There are 27 trees
on each tropical cubic surface, attached to a bounded complex with up to 73
polygons. The maximal cones in the 4-dimensional moduli fan reveal two generic
types of such surfaces.
