Horton Self-similarity of Kingman’s Coalescent Tree

The paper establishes Horton self-similarity for a tree representation of Kingman’s coalescent process. The proof is based on a Smoluchowski-type system of ordinary differential equations that describes evolution of the number of branches of a given HortonStrahler order in a tree that represents Kingman’s N -coalescent, in a hydrodynamic limit. We also demonstrate a close connection between the combinatorial Kingman’s tree and the combinatorial level set tree of a white noise, which implies Horton self-similarity for the latter.