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