Simply Generated Unrooted Plane Trees

TL;DR

This paper demonstrates that random unrooted plane trees with degree-based weights can be approximated by conditioned Galton--Watson trees, enabling the transfer of many known results from the rooted case to unrooted trees.

Contribution

It establishes a geometric approximation of weighted unrooted plane trees by Galton--Watson trees, extending known results from rooted trees to the unrooted case.

Findings

Approximation by Galton--Watson trees under positive radius of convergence.

Results on scaling limits, diameter bounds, and degree distribution for unrooted trees.

Extension of planted tree results to unrooted plane trees.

Abstract

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.