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.