  We study tropically planar graphs, which are the graphs that appear in smooth
tropical plane curves. We develop necessary conditions for graphs to be
tropically planar, and compute the number of tropically planar graphs up to
genus $7$. We provide non-trivial upper and lower bounds on the number of
tropically planar graphs, and prove that asymptotically $0\%$ of connected
trivalent planar graphs are tropically planar.
