Human introduction of species around the world is not exhaustive. Comparing characteristics of non-native species that have been introduced with characteristics of non-native species that could be introduced can reveal biases in which species were introduced in the past, and can highlight future threats. We test for introduction biases in non-native trees introduced to Australia using a dataset of 57,958 tree species known globally. We found that: (1) non-native species were selectively introduced, and do not form a random subset of all non-native species, though (2) they do represent the functional trait-space occupied by tree species globally; (3) trees naturalised in numerous countries and introduced for more uses are more likely to be introduced in Australia; and (4) introduction debt can indicate characteristics of potential future non-native introductions. For Australia, tree species from the tropics and global south pose an especially high future invasion risk.