  We show that each orbit of a Borel subgroup $B_{n-1}$ of GL(n-1)
(respectively SO(n-1)) on the flag variety of GL(n) (respectively of SO(n)) is
a bundle over a $B_{n-1}$-orbit on a generalized flag variety of GL(n-1)
(respectively SO(n-1)), with fiber isomorphic to an orbit of an analogous
subgroup on a smaller flag variety. As a consequence, we develop an inductive
procedure to classify $B_{n-1}$-orbits on the flag variety. Our method is
essentially uniform in the two cases. As further consequences, in the sequel to
this paper we give an explicit combinatorial classification of orbits and
determine completely the closure relation between orbit closures. This further
develops work of Hashimoto in the general linear group case.
