A weed is a plant out of place or not desired in the given context. By that definition, a tree can be a weed if it is planted in the wrong place, if it seeds on its own in an undesirable location, or if it is a species that has the potential to be invasive. Invasive trees, once established, tend to reproduce rapidly and out-compete other trees and plants.
You can check with your local Extension office for help determining whether a particular tree might be considered a weed or an invasive species in your region. Another good resource is http://www.invasivespeciesinfo.gov/.