The purpose of this research study is to learn about the medical effects, safety, and how the Walnut Oral Immunotherapy (OIT) treatment affects your body (immune system). This type of immunotherapy involves giving increasing doses of walnut allergen to gradually build up a person's tolerance to walnut and at least one other tree nut. The goal of the study is to determine whether participants can tolerate (eat) walnuts and at least one other tree nut in their diet after stopping the study therapy.