  We study random unrooted plane trees with $n$ vertices sampled according to
the weights corresponding to the vertex-degrees. Our main result shows that if
the generating series of the weights has positive radius of convergence, then
this model of random trees may be approximated geometrically by a
Galton--Watson tree conditioned on having a large random size.
  This implies that a variety of results for the well-studied planted case also
hold for unrooted trees, including Gromov--Hausdorff--Prokhorov scaling limits,
tail-bounds for the diameter, distributional graph limits, and limits for the
maximum degree. Our work complements results by Wang~(2016), who studied random
unrooted plane trees whose diameter tends to infinity.
