Adaptive enrichment designs involve preplanned rules for modifying patient enrollment criteria based on data accrued in an ongoing trial. These designs may be useful when it is suspected that a subpopulation, e.g., defined by a biomarker or risk score measured at baseline, may benefit more from treatment than the complementary subpopulation. We compare two types of such designs, for the case of two subpopulations that partition the overall population. The first type starts by enrolling the subpopulation where it is suspected the new treatment is most likely to work, and then may expand inclusion criteria if there is early evidence of a treatment benefit. The second type starts by enrolling from the overall population and then may selectively restrict enrollment if sufficient evidence accrues that the treatment is not benefiting a subpopulation. We construct designs of each type that guarantee strong control of the familywise Type I error rate. We then compare performance of the designs from each type under different scenarios; the scenarios mimic key features of a completed non-inferiority trial of HIV treatments. Performance criteria include power, sample size, duration, and number of participants assigned to a highly inferior treatment.