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Knuth, Donald E. (1971), "Optimum binary search trees", Acta Informatica 1 (1): 14–25,doi:10.1007/BF00264289

Please have a look at this paper, specifically page 18 in which he tries to prove his lemma that $R_{0,n-1} \leq R_{0,n} $ here $R$ refers to the minimal optimal node which will be the root of the binary tree containing elements $a_{0} ... a_{n} $ .

I understood the idea of the proof using induction that for some $k \, j_{k}=i_{k} $ . Now the next part of the proof is cutting and replacing,I have understood perfectly till there. What i don't understand is how $F''$ weighted path length is equal to weighted path length of $F'$ for all $a_{n}$. Can anyone please give me a hint or a solution to that.

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