Consecutive Pattern Avoidances in Non-crossing Trees

In this paper, motivated by the concepts of increasing (alternating) non-crossing trees, the problem of consecutive pattern avoidances in non-crossing trees is proposed. Some given patterns of length two and three are investigated in detail. The Lagrange inversion formula is used to obtain the explicit formulas for these cases. Bijections are established between non-crossing trees avoiding special patterns and Schröder paths.