From Gentzen to Jaskowski and Back: Algorithmic Translation of Derivations Between the Two Main Systems of Natural Deduction - von Plato, Jan (Bulletin of the Section of Logic, 2017)
The way from linearly written derivations in natural deduction, introduced by Jaskowski and often used in textbooks, is a straightforward root-first translation. The other direction, instead, is tricky, because of the partially ordered assumption formulas in a tree that can get closed by the end of a derivation. An algorithm is defined that operates alternatively from the leaves and root of a derivation and solves the problem.