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., T A(n) ≤ T 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.2000 AMS Classification: 06A12, 06F05, 16B50.
The theme of this book is the Structure Theory of compact groups. It contains a completely selfcontained introduction to linear Lie groups and a substantial body of material on compact Lie groups. The authors' approach is distinctive in so far as they define a linear Lie group as a particular subgroup of the multiplicative group of a Banach algebra. Compact Lie groups are recognized at an early stage as being linear Lie groups. This approach avoids the use of machinery on manifolds. The text is written in a style to make it accessible to the beginning graduate student with a basic knowledge in analysis, algebra, and topology. At the same time the expert will find it an excellent and rich source of information on the general structure theory of compact groups.
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