Cyclic sieving phenomena for trees and tree-rooted maps

Abstract: We prove cyclic sieving phenomena satisfied by corner-rooted plane trees (alias ordered trees). The sets of rooted plane trees that we consider are: (1) all trees with $n$ nodes; (2) all trees with $n$ nodes and $k$ leaves; (3) all trees with a given degree distribution of the nodes. Moreover, we consider four different cyclic group actions: (1) the root is moved to the next corner along a tour of the tree; (2) only trees in which the root is at a leaf are considered, and the action moves the root to the next leaf; (3) only trees in which the root is at a non-leaf are considered, and the action moves the root to the next non-leaf corner; (4) only trees in which the root is at a node of degree $δ$ are considered, for a fixed $δ$, and the action moves the root to the next corner of this type. We prove a cyclic sieving phenomenon for each meaningful combination of these sets and actions. As a bonus, we also establish corresponding cyclic sieving phenomena for tree-rooted planar maps.