The number of DFAs for a given spanning tree
 P. Babaali,
 E. CartaGerardino,
 C. Knaplund
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In the last few decades, several techniques to randomly generate a deterministic finite automaton have been developed. These techniques have implications in the enumeration and random generation of automata of size n. One of the ways to generate a finite automaton is to generate a random tree and to complete it to a deterministic finite automaton, assuming that the tree will be the automaton’s breadthfirst spanning tree. In this paper we explore further this method, and the string representation of a tree, and use it to counting the number of automata having a tree as a breadthfirst spanning subtrees. We introduce the notions of characteristic and difference sequence of a tree, and define the weight of a tree. We also present a recursive formula for this quantity in terms of the “derivative” of a tree. Finally, we analyze the implications of this formula in terms of exploring trees with the largest and smallest number of automata in the span of the tree and present simple applications for finding densities of subsets of DFAs.
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 Title
 The number of DFAs for a given spanning tree
 Journal

The Journal of Supercomputing
Volume 65, Issue 2 , pp 710722
 Cover Date
 20130801
 DOI
 10.1007/s1122701309570
 Print ISSN
 09208542
 Online ISSN
 15730484
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Random generation of automata
 Spanning trees
 Tree derivative
 Characteristic of a tree
 Weight of a tree
 Distribution of automata
 Industry Sectors
 Authors

 P. Babaali ^{(1)}
 E. CartaGerardino ^{(1)}
 C. Knaplund ^{(2)}
 Author Affiliations

 1. Department of Mathematics and Computer Science, York College, City University of New York, 9420 Guy R. Brewer Blvd., Jamaica, NY, 11415, USA
 2. Department of Mathematics, Graduate Center of the City University of New York, 365 5th Avenue, New York, NY, USA