  We address the problem of existence of refined (i.e., depending on a formal
parameter) tropical enumerative invariants, and we present two new examples of
a refined count of rational marked tropical curves. One of the new invariants
counts plane rational tropical curves with an unmarked vertex of arbitrary
valency. It was motivated by the tropical enumeration of plane cuspidal
tropical curves given by Y. Ganor and the author, which naturally led to
consideration of plane tropical curves with an unmarked four-valent vertex.
Another refined invariant counts rational tropical curves of a given degree in
the Euclidean space of arbitrary dimension matching specific constraints, which
make the spacial refined invariant similar to known planar invariants.
