What is the smallest possible depth of a leaf in a decision tree for a comparison sort?
The smallest possible depth is \(n - 1\).
Think about the best-case scenario for a comparison sort. Imagine the input array is already sorted. Even in this ideal case, we still need to verify that it’s sorted by making comparisons. Consider how few comparisons we can get away with.
The most efficient way to verify a sorted array is to compare adjacent elements. For an array of length \(n\), we need at least \(n - 1\) comparisons to confirm that each element is less than or equal to the next one. This is the absolute minimum because we need to establish relationships between all consecutive pairs.
For example, with 3 elements, we need at least 2 comparisons. We might compare \(a_1\) with \(a_2\) and \(a_2\) with \(a_3\). If both comparisons show the elements are in order, we’ve confirmed the array is sorted.
This best case corresponds to the shallowest leaf in the decision tree. The path from root to this leaf represents the minimum number of comparisons needed for the most favorable input permutation.