  A rooted tree T is a poset whose Hasse diagram is a graph-theoretic tree
having a unique minimal element. We study rowmotion on antichains and lower
order ideals of T. Recently Elizalde, Roby, Plante and Sagan considered
rowmotion on fences which are posets whose Hasse diagram is a path (but
permitting any number of minimal elements). They showed that in this case, the
orbits could be described in terms of tilings of a cylinder. They also defined
a new notion called homometry which means that a statistic takes a constant
value on all orbits of the same size. This is a weaker condition than the
well-studied concept of homomesy which requires a constant value for the
average of the statistic over all orbits. Rowmotion on fences is often
homometric for certain statistics, but not homomesic. We introduce a tiling
model for rowmotion on rooted trees. We use it to study various specific types
of trees and show that they exhibit homometry, although not homomesy, for
certain statistics.
