Abstract
We introduce a new lattice structure B n on binary trees of size n . We exhibit efficient algorithms for computing meet and join of two binary trees and give several properties of this lattice. More precisely, we prove that the length of a longest (resp. shortest) path between 0 and 1 in B n equals to the Eulerian numbers 2 n − ( n + 1 ) (resp. ( n − 1 ) 2 ) and that the number of coverings is ( 2 n n − 1 ) . Finally, we exhibit a matching in a constructive way. Then we propose some open problems about this new structure.
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