The existence of rational plane curves of a given degree with prescribed singularities is a subtle and active area in algebraic geometry. This problem turns out to be closely related to difficult enumerative problems which arise in symplectic field theory, which in turn play a central role in the theory of high dimensional symplectic embeddings. In this talk, I will discuss various perspectives on these enumerative problems and present a new closed formula for relevant curve counts as a sum over decorated trees.