Recently, the authors introduced new families of Dyck paths having a first decomposition constrained by the height or by the number of returns. In this work we extend the study to Motzkin paths and 2-colored Motzkin paths. For these new sets, we provide enumerative results by giving bivariate generating functions with respect to the length and another parameter, and we construct one-to-one correspondences with several restricted classes of ordered trees. We also deal with Schr¨oder and Riordan paths. As a byproduct, we present a bijective proof of M(x)2 = 1 1?2xM( x2 1?2x), where M(x) is the generating function of Motzkin numbers.
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