Brill-Noether theory aims to study degree d rational maps from a fixed 
algebraic curve X to a projective space P^r. Here d and r are parameters, 
and a typical game is to fix one and attempt minimizing the other. 
For r=1 the minimal d is called the gonality of X.
Big progress was made in the 70s and 80s by parametrising with a moduli 
space and studying its properties, such as dimension and connectedness.
Tropical geometry introduces polyhedral complexes as combinatorial 
analogues of classical algebraic varieties. Many problems in algebraic 
geometry have a tropical counterpart, and there tends to be an interplay 
which sheds light on both sides of the story.
We study a tropical version of gonality, where we look at tropical morphisms 
from metric graphs to metric trees, and construct a moduli space which 
closely resembles, in spirit and nature of the results obtained, the 
classical story. One striking result is that the number of such morphisms 
for fixed even genus metric graph is a Catalan number.
Our methods are entirely combinatorial and, moreover, constructive.