Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)
On trees, tanglegrams, and tangled chains
Sara Billey  1  , Matjaz Konvalinka  2  , Frderick Matsen Iv  3  
1 : Department of Mathematics [Seattle]
University of Washington Department of Mathematics Box 354350 Seattle, WA 98195-4350 -  États-Unis
2 : Departement of Mathematics [Slovenia]  -  Website
University of Ljubljana, Faculty of Mathematics and Physics, Jadranska 19, 1000 Ljubljana, Slovenia -  Slovénie
3 : Computational Biology Program  -  Website
Division of Public Health Sciences, Fairview Avenue North, Seattle, WA 98109 -  États-Unis

Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared. 

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