In the lower proof, we saw that the number of leaves had to be at least...

 

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In the lower proof, we saw that the number of leaves had to be at least N! to sort every possible input (permutation). If the decision tree contained more than N! leaves, what would that mean? The decision tree cannot be a binary tree, but must be a tree with three children per node. The decision tree contains multiple paths to sorting the same input. The decision tree contains incorrect paths, meaning the algorithm cannot sort properly. The decision tree cannot be for a sorting algorithm. In the lower proof, we saw that the number of leaves had to be at least N! to sort every possible input (permutation). If the decision tree contained more than N! leaves, what would that mean? The decision tree cannot be a binary tree, but must be a tree with three children per node. The decision tree contains multiple paths to sorting the same input. The decision tree contains incorrect paths, meaning the algorithm cannot sort properly. The decision tree cannot be for a sorting algorithm.

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