Sentence:  A natural deduction tree can be put in exact correspondence with typed &lambda;-calculus (that can be considered as just a notation for such deductions), and the cut-elimination in Natural Deduction corresponds exactly to <WSD>&beta;-reduction</WSD> in &lambda;-calculus. 
Meaning of the word marked with <WSD> tags:  (computing theory) One of the three rewrite rules of the lambda calculus, which states that the application of a lambda abstraction (\lambda x. t)  to a term s yields an expression t[s/x], ""t" with "s" instead of "x"", where all free instances of "x" in "t" have been replaced by "s".