Enumeration of labeled trees and Dyck tilings

We study a partially ordered set of planar labeled rooted trees by use of combinatorial objects called Dyck tilings. A generating function of the poset is factorized when the minimum element of the poset is $312$-avoiding and satisfies some extra condition. We define a cover relation on rational Dyck tilings by that of labeled trees, and show that increasing and decreasing labelings are dual to each other. We consider two decompositions of a rational $(a,b)$-Dyck tiling: one is into $ab$ Dyck tilings and the other is into a $(1,b)$-Dyck tilings. In the first case, we show that the weight of the $(a,b)$-Dyck tiling is the sum of the weights of $ab$ Dyck tilings. In the second case, we introduce a cover relation on $(1,b)$-Dyck tilings and obtain a poset of $(a,b)$-Dyck tilings by this decomposition.