The Register Function and Reductions of Binary Trees and Lattice Paths

TL;DR

This paper introduces a new reduction-based interpretation of the register function for binary trees and lattice paths, analyzing their asymptotic behavior and complexity measures.

Contribution

It presents a novel reduction procedure linking the register function to tree and lattice path complexity, with detailed asymptotic analysis.

Findings

Asymptotic behavior of the number of branches in reduced trees

Complexity measures for lattice paths derived from reductions

Quantitative analysis of reduced path sizes

Abstract

The register function (or Horton-Strahler number) of a binary tree is a well-known combinatorial parameter. We study a reduction procedure for binary trees which offers a new interpretation for the register function as the maximal number of reductions that can be applied to a given tree. In particular, the precise asymptotic behavior of the number of certain substructures ("branches") that occur when reducing a tree repeatedly is determined. In the same manner we introduce a reduction for simple two-dimensional lattice paths from which a complexity measure similar to the register function can be derived. We analyze this quantity, as well as the (cumulative) size of an (iteratively) reduced lattice path asymptotically.