To solve this problem, we need to identify the organism that forms nitrogen-fixing nodules on the roots of $Alnus. \newline$Alnus is a genus of trees commonly known as alders, which are known to form symbiotic relationships with nitrogen-fixing bacteria.$\newline$Let's analyze the options:$\newline$• Option 1: $Rhizobium$ is a genus of bacteria that forms nodules on the roots of legumes, such as peas and beans,$\newline$ but not on $Alnus. \newline$• Option 2: $Frankia$ is a genus of actinobacteria that forms nitrogen-fixing nodules on the roots of non-leguminous plants,$\newline$ including $Alnus. \newline$• Option 3: $Rhodospirillum$ is a genus of purple non-sulfur bacteria that does not form nodules on plant roots.$\newline$• Option 4: $Beijerinckia$ is a genus of free-living nitrogen-fixing bacteria that does not form nodules on plant roots.$\newline$Based on this analysis, the correct answer is Option 2: $Frankia. \newline$Frankia forms symbiotic relationships with $Alnus$ and produces nitrogen-fixing nodules on its roots.