The Register Function and Reductions of Binary Trees and Lattice Paths
Combinatorics
2016-05-12 v2
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.
Keywords
Cite
@article{arxiv.1602.06200,
title = {The Register Function and Reductions of Binary Trees and Lattice Paths},
author = {Benjamin Hackl and Clemens Heuberger and Helmut Prodinger},
journal= {arXiv preprint arXiv:1602.06200},
year = {2016}
}
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Extended abstract