Assume we insert sequences of English letters into an empty 2-3-tree, following the standard alpha-betic ordering when making comparisons: A<B<C< ... < Z.
(a) (3 points) Assume you insert the letters R, E, L, A, T, I, O, N, S into a fresh 2-3-tree T in that exact order. Draw the resulting tree. What is its height?
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(b) (4 points) Let h be your answer to part (a). Is it possible to get less than h by inserting the letters from (a) in a different order? If not, prove it. If yes, give the best ordering you can. What is the depth $h_{min}$ you get?
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(c) (3 points) Can you achieve $h_{min}$ with lexicographic order A, E, I, L, N, O, R, S, T? Draw the resulting tree.
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(d) (4 points) Extra credit: Can you achieve $h_{min}$ for part (b) using an English word that is a permutation of R, E, L, A, T, I, O, N, S?
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(e) (4 points) Give an example of a 2-3-tree T with exactly 5 leaves and two distinct values a, b, such that one gets different final trees if one first inserts a into T and then deletes b, instead of first deleting b from T and then inserting a. Explain what happened.
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