Invasive Species Prevention and Forest Restoration Act This bill directs the Department of Agriculture (USDA) to award competitive grants to certain institutions for research to promote the restoration of tree species affected by non-native plant pests and noxious weeds. USDA may also award competitive grants to implement the research or other solutions to restore forest tree species native to the United States that have suffered severe levels of mortality caused by non-native plant pests and noxious weeds. The bill also expands the authority of USDA to transfer funds to respond to an emergency involving an outbreak of a plant pest or noxious weed. Further, USDA must enter into an agreement with the National Academy of Sciences or another nongovernmental entity to analyze available resources in federal agencies for research of, and solutions to, non-native forest pests and pathogens.