  In this paper, we introduce tree varieties as a natural generalization of
products of partial flag varieties. We study orbits of the PGL action on tree
varieties. We characterize tree varieties with finitely many PGL orbits,
generalizing a celebrated theorem of Magyar, Weyman and Zelevinsky. We give
criteria that guarantee that a tree variety has a dense PGL orbit and provide
many examples of tree varieties that do not have dense PGL orbits. We show that
a triple of two-step flag varieties $F(k_1, k_2; n)^3$ has a dense PGL orbit if
and only if $k_1 + k_2 \not= n$.
