There are 2 type of picnic areas: 1) Reservation only groves and 2) Open groves. For Open groves you don't need any permit (alcohol too is prohibited in Open groves). You just need to occupy the tables (as many as you want) as early as possible. For Reservation groves, you can contact the Parks and Recreation office and reserve your spot. You can also apply for beer permit. Important Tip: If a reservation grove is not reserved on a particular day, it is considered open to all. You can inquire with park rangers in the morning and they will tell you which of the reservation groves have no reservation for that day. You can then use those groves as Open groves.
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