  This paper starts with an observation that two infinite series of simplicial
complexes, which a priori do not seem to have anything to do with each other,
have the same homotopy type. One series consists of the complexes of directed
forests on a double directed string, while the other one consists of
Shapiro-Welker models for the spaces of hyperbolic polynomials with a triple
root. We explain this coincidence in the more general context by finding an
explicit homotopy equivalence between complexes of directed forests on a double
directed tree, and doubly disconnecting complexes of a tree.
