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From Gentzen to Jaskowski and Back: Algorithmic Translation of Derivations Between the Two Main Systems of Natural Deduction
Bulletin of the Section of Logic, Tome 46 (2017) no. 1-2.

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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.
Keywords: proof systems, linear natural deduction, Gentzen, Jaśkowski 
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     author = {von Plato, Jan},
     title = {From {Gentzen} to {Jaskowski} and {Back:} {Algorithmic} {Translation} of {Derivations} {Between} the {Two} {Main} {Systems} of {Natural} {Deduction}},
     journal = {Bulletin of the Section of Logic},
     publisher = {mathdoc},
     volume = {46},
     number = {1-2},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BSL_2017_46_1-2_a2/}
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von Plato, Jan. From Gentzen to Jaskowski and Back: Algorithmic Translation of Derivations Between the Two Main Systems of Natural Deduction. Bulletin of the Section of Logic, Tome 46 (2017) no. 1-2. http://geodesic.mathdoc.fr/item/BSL_2017_46_1-2_a2/