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We study one-sided substitution subshifts, and how they can be representedusing Bratteli-Vershik systems. In particular we focus on minimal recognizablesubstitutions such that the generated one-sided substitution subshift containsonly one non-shift-invertible element (a branch point), and we call thesesubstitutions quasi-invertible. We give an algorithm to check whether asubstitution is quasi-invertible, and show that any substitution with arational Perron value is orbit equivalent to a quasi-invertible substitution.If the quasi-invertible substitution is left proper, then its subshift is equalto a substitution subshift where the original branch point is the newsubstitution fixed point. We use these results to prove that any reasonablequasi-invertible substitution subshift has a Bratteli-Vershik representation.We also give an example of a pair of substitutions whose 2-sided subshifts aretopologically conjugate, while their 1-sided subshifts are not.

Bratteli-Vershik representations of some one-sided substitution subshifts