Reassortment among influenza A viruses (IAV) facilitates genomic evolution and has been associated with interspecies transmission and pandemics. We introduce a novel tool called TreeSort that accurately identifies recent and ancestral reassortment events on datasets with thousands of IAV whole genomes. The algorithm has immediate relevance to modern large-scale surveillance studies that generate thousands of whole genomes. TreeSort uses the phylogeny of a selected IAV segment as a reference and finds the branches on the phylogeny where reassortment has occurred with high probability by incorporating information from other gene segments. The tool reports the particular gene segments that were involved in reassortment and how different they are from prior gene pairings. Using TreeSort, we studied reassortment patterns of different IAV subtypes isolated in avian, swine, and human hosts. Avian IAV demonstrated more reassortment than human and swine IAV, with the avian H7 subtype displaying the most frequent reassortment. Reassortment in the swine and human H3 subtypes was more frequent than in the swine and human H1 subtypes, respectively. The highly pathogenic avian influenza H5N1 clade 2.3.4.4b had elevated reassortment rates in the 2020-2023 period; however, the surface protein-encoding genes (HA, NA, and MP) co-evolved together with almost no reassortment among these genes. We observed similar co-evolutionary patterns with very low rates of reassortment among the surface proteins for the human H1 and H3 lineages, suggesting that strong co-evolution and preferential pairings among surface proteins are a consequence of high viral fitness. Our algorithm enables real-time tracking of IAV reassortment within and across different hosts, can be applied to determine how specific genes evolve in response to reassortment events, and can identify novel viruses for pandemic risk assessment. TreeSort is available at https://github.com/flu-crew/TreeSort.