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We compare the average time complexity of the breadth-first search (BFS) and depth-first search (DFS) algorithms, when the target node is selected uniformly at random among all nodes at level $\\ell$ in the ordered trees with $n$ edges. Intuition suggests that BFS should have better average performance when $\\ell$ is small, while DFS must have an advantage when $\\ell$ is large. But where exactly is the threshold, as a function of $n$, and is it unique? 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