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The Algebra of Binary Search Trees

F. Hivert·J. -C. Novelli·J. -Y. Thibon
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Abstract

We introduce a monoid structure on the set of binary search trees, by a process very similar to the construction of the plactic monoid, the Robinson-Schensted insertion being replaced by the binary search tree insertion. This leads to a new construction of the algebra of Planar Binary Trees of Loday-Ronco, defining it in the same way as Non-Commutative Symmetric Functions and Free Symmetric Functions. We briefly explain how the main known properties of the Loday-Ronco algebra can be described and proved with this combinatorial point of view, and then discuss it from a representation theoretical point of view, which in turns leads to new combinatorial properties of binary trees.

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F. Hivert, J. -C. Novelli, J. -Y. Thibon. 2004-01-09. The Algebra of Binary Search Trees. https://arxiv.org/abs/math/0401089

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