When N processors perform depth-first search on disjoint parts of a state space tree to find a solution, the speedup can be superlinear (i.e., > N) or sublinear (i.e., <N) depending upon when a solution is first encountered in the space by one of the processors. It may appear that on the average, the speedup would be either linear or sublinear. Using an analytical model, we show that if the search space has more than one solution and if these solutions are randomly distributed in a relatively small region of the search space, then the average speedup in parallel depth-first search can be superlinear. If all the solutions (one or more) are uniformly distributed over the whole search space, then the average speedup is linear. This model is validated by our experiments on synthetic state-space trees and the 15-puzzle problem. The same model predicts average superlinear speedup in parallel best-first branch-and-bound algorithms on suitable problems.