  Floor diagrams are a class of weighted oriented graphs introduced by E.
Brugalle and the second author. Tropical geometry arguments lead to
combinatorial descriptions of (ordinary and relative) Gromov-Witten invariants
of projective spaces in terms of floor diagrams and their generalizations. In a
number of cases, these descriptions can be used to obtain explicit (direct or
recursive) formulas for the corresponding enumerative invariants. In
particular, we use this approach to enumerate rational curves of given degree
passing through a collection of points on the complex plane and having maximal
tangency to a given line. Another application of the combinatorial approach is
a proof of a conjecture by P. Di Francesco -- C. Itzykson and L. Goettsche that
in the case of a fixed cogenus, the number of plane curves of degree d passing
through suitably many generic points is given by a polynomial in d, assuming
that d is sufficiently large. Furthermore, the proof provides a method for
computing these "node polynomials."
  A labeled floor diagram is obtained by labeling the vertices of a floor
diagram by the integers 1,...,d in a manner compatible with the orientation. We
show that labeled floor diagrams of genus 0 are equinumerous to labeled trees,
and therefore counted by the celebrated Cayley's formula. The corresponding
bijections lead to interpretations of the Kontsevich numbers (the genus-0
Gromov-Witten invariants of the projective plane) in terms of certain
statistics on trees.
