  We prove that the complement of a $\sigma$-compact subset of a topological
space that has a $\pi$-tree also has a $\pi$-tree. To do this, we construct the
foliage hybrid operation, which deals with foliage trees (that is,
set-theoretic trees with a `leaf' at each node). Then using this operation we
modify a $\pi$-tree of a space and get a $\pi$-tree for its subspace.
