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

The Register Function and Reductions of Binary Trees and Lattice Paths