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Parking functions, classically defined in terms of cars with preferredparking spots on a directed path attempting to park there, arise in manycombinatorial situations and have seen various generalizations. In particular,parking functions have been defined for general digraphs, which yields manymore enumeration problems. For example, in a directed tree whose edges areorientated away from the root, it is unknown in general how the number ofparking functions on it changes once the orientation is reversed, even in thecase when the tree is a star. We show that this orientation reversal results inmore parking functions on the directed star in most cases, after which weextend these methods to show that this also results in more parking functionson the general directed tree if, in some sense, the number of vertices greatlyexceeds the number of cars.

Parking functions on directed stars and orientation reversal with an extension to general directed trees