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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.

Directed Trees in a String, Real Polynomials with Triple Roots, and Chain Mails