Topological and categorical properties of binary trees

H. Pajoohesh

doi:10.4995/agt.2008.1863

Authors

H. Pajoohesh Medgar Evers College

DOI:

https://doi.org/10.4995/agt.2008.1863

Keywords:

Algorithm, Decision tree, Lower topology, Semilattice

Abstract

Binary trees are very useful tools in computer science for estimating the running time of so-called comparison based algorithms, algorithms in which every action is ultimately based on a prior comparison between two elements. For two given algorithms A and B where the decision tree of A is more balanced than that of B, it is known that the average and worst case times of A will be better than those of B, i.e., â‚¸_A(n) ≤â‚¸_B(n) and T^W_A (n)≤T^W_B (n). Thus the most balanced and the most imbalanced binary trees play a main role. Here we consider them as semilattices and characterize the most balanced and the most imbalanced binary trees by topological and categorical properties. Also we define the composition of binary trees as a commutative binary operation, *, such that for binary trees A and B, A * B is the binary tree obtained by attaching a copy of B to any leaf of A. We show that (T,*) is a commutative po-monoid and investigate its properties.

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Author Biography

H. Pajoohesh, Medgar Evers College

Department of Mathematics, 1150 Carroll Street, Brooklyn, NY 11225-2298, USA.

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