  Cover-inclusive Dyck tilings are tilings of skew Young diagrams with ribbon
tiles shaped like Dyck paths, in which tiles are no larger than the tiles they
cover. These tilings arise in the study of certain statistical physics models
and also Kazhdan--Lusztig polynomials. We give two bijections between
cover-inclusive Dyck tilings and linear extensions of tree posets. The first
bijection maps the statistic (area + tiles)/2 to inversions of the linear
extension, and the second bijection maps the "discrepancy" between the upper
and lower boundary of the tiling to descents of the linear extension.
