Asymptotics of the counting function of lengths corresponding to classes of equivalent routes for an infinite binary metric tree

M. V. Vakhitov^a, D. S. Minenkov^b, A. A. Tolchennikov^b, V. L. Chernyshev<sup>c

a</sup> Lomonosov Moscow State University
^b Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow
^c National Research University Higher School of Economics, Moscow

Abstract: Consider routes on a binary metric tree. Routes that differ only in the order in which they traverse edges are grouped into classes of equivalent routes. An explicit combinatorial formula is found for the counting function of the lengths corresponding to the classes of equivalent routes. For the case where edge lengths grow asymptotically linearly, an estimate for the logarithm of the counting function for the lengths of the classes of equivalent routes is obtained, as well as for the lengths themselves.

Keywords: routes on a metric graph, binary tree, abstract primes, counting functions, number of partial partitions.

UDC: 519.173

Received: 15.10.2024

DOI: 10.4213/dm1851