Elsevier

Theoretical Computer Science

Volume 705, 1 January 2018, Pages 31-57
Theoretical Computer Science

Reductions of binary trees and lattice paths induced by the register function

https://doi.org/10.1016/j.tcs.2017.09.015Get rights and content
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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

Register function
Binary tree
Lattice path
Asymptotics

Cited by (0)

This is the full version of the extended abstract [12].

1

B. Hackl and C. Heuberger are supported by the Austrian Science Fund (FWF): P 24644-N26 and by the Karl Popper Kolleg “Modeling-Simulation-Optimization” funded by the Alpen-Adria-Universität Klagenfurt and by the Carinthian Economic Promotion Fund (KWF).

2

H. Prodinger is supported by incentive grant 28110 of the National Research Foundation of South Africa.