We can represent the comparisons made along a root-to-leaf path in the decision tree as a directed graph. Each node represents an element of the array ๐ด[๐]*A*[*I*]. An edge from node ๐* *to node ๐ย indicates a comparison between ๐ด[๐]*A*[*i*]ย and ๐ด[๐]*A*[*j*].

For the permutation at a leaf to be correctly determined, the comparison graph must be connected. This means there must be a path of comparisons connecting every pair of nodes. If the graph is not connected, there will be at least two disjoint subsets of elements whose relative order hasn't been determined.

In an undirected graph, a tree with ๐ย nodes has ๐โ1ย edges and is connected. Similarly, in a directed graph representing comparisons, you need at least ๐โย comparisons to ensure connectivity. Therefore, with fewer than ๐โ1ย comparisons, you cannot guarantee that the graph is connected, meaning there would be some relative orders of elements that are not determined.

Hence, the smallest possible depth of a leaf in a decision tree, where the permutation of elements at that leaf is guaranteed to be correct, is indeed ๐โ1. This means that in the best case, you would need at least ๐โ1ย comparisons to fully determine the correct order of ๐ย elements. This depth would correspond to a scenario where the algorithm encounters a highly favorable input sequence, such as an already sorted array in some comparison sorts (e.g., insertion sort). However, this is not the average or worst-case depth, which for comparison-based sorting algorithms is generally ๐(๐logโก๐).

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