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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-Prokhorovdoi:10.30757/alea.v16-12 fatcat:ktf3r74oqjcabg6yt4fdogriyq