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.
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).