Zero-nonzero tree patterns that allow S_n^∗

For (Formula presented.), let (Formula presented.). A zero-nonzero pattern (Formula presented.) of order n allows (Formula presented.) if (Formula presented.), the inertia set of (Formula presented.), and requires (Formula presented.) if (Formula presented.). The study of zero-nonzero patterns allowing (Formula presented.) was introduced by Berliner et al. [Inertia sets allowed by matrix patterns. Electron J Linear Algebra. 2018;34:343–355]. In this paper, we give characterizations of irreducible zero-nonzero path patterns and double-star patterns that allow (Formula presented.). Moreover, all irreducible zero-nonzero tree patterns of order 5 and 6 that allow (Formula presented.) are also characterized.