Ask Mr. Smarty Plants

There are mimosas and mimosas. Two native trees that are called mimosas are Mimosa texana (Texas mimosa) and Desmanthus illinoensis (Illinois bundleflower). However, we're betting the one you refer to is Albizia julibrissi, a non-native of North America, instead native to an area ranging from Iran to Japan. All three are in the Fabaceae or pea family,but there the resemblance ends. At the Lady Bird Johnson Wildflower Center, we promote the use of plants native to North America because they need less fertilizer, water and maintenance growing in their natural habitats, and thus are better for the environment. The problem we had in trying to find the answer to your question is that most websites were more interested in getting rid of the mimosa than getting it to grow normally.

Plant Conservation Alliance Least Wanted Silk Tree.

USDA Forest Service Albizia julibrissin. The USDA Forest Service is usually at least neutral about non-natives but it doesn't have much good to say, and no information on how to make a single trunk branch.

But you didn't ask us to evaluate the value of your tree; you asked how to make it branch. Ordinarily, we would recommend pruning just below a bud area to promote growth of new branches. However, what you have now is a single leader, and most experts strongly advise against pruning leaders. You shouldn't even consider pruning now, in the heat of the summer, but wait until winter has cleaned off the foliage and permitted you to see the skeleton of the tree. By then, perhaps you will be able to see a branch radiating out from the leader that would make an acceptable crotch for the tree. This article from the University of Minnesota Extension on Pruning Trees and Shrubs seems to us to have the best diagrams and instructions for dealing with a young tree. You need to be aware that the mimosa is basically a weak tree, and a storm or even a strong wind could snap that thin leader in two. If that happens, we would advise you to start over with a stronger, native tree.