Here we just use what is growing around us, either species native to the area or ones planted.
My best advice is to grow own that will be best suited for your area climate and of a size that is manageable for your land. Some species can grow to really huge heights. And you would want one that sets down a good tap root, as some are quite shallow rooted and can come down in heavy rain and winds. A couple that I am not sure about the birds liking would be the Alpine and the Tasmanian. They are more cold climate trees and budgies in the wild budgies live in semi desert.
One that you could try to get hold of is a Red Flowering Gum (Eucalyptus ficifolia) or one of the hybrids. The hybrids don't grow as tall and are very slow growing and have been specifically developed for the home garden. They produces masses of red or musk pink flowers and are very attractive. My neighbour has one in their yard and it is really pretty when in flower. While our native lorikeets love the flowers of them, I think many of your nectar feeding birds would love them as well. Here is a couple of links about them.
Red-flowering Gums - Burke's Backyard
As a bit of side information Don Burke from Burke's Backyard is a budgie breeder.