There are two kinds of trees that are called mimosa. From what I've heard, your mimosa with the yellow is the "true" mimosa (whatever that means). What we call mimosa in the U.S. is also called silk tree. I personally have never heard anyone use that name.
The tree I have in my sideyarden that I was told is a true silk tree has no blooms and from I have been told never will. Why I do not know. This is supposedly the tree brought from China to try to start a silk industry in this country but the silk worm responsible ro making the webs to make the fabric didn't seem to like this country's climate and they all died.
The tree that silk worms feed off of is a mulberry tree. If I remember correctly, most mulberry trees planted nowdays are of a variety that does not bloom because so many people are allergic to the pollen.
I could be wrong, but I think mulberries might be male and female on separate plants. Apparently they were popular in the 1960s and there are a lot of them lining streets around here. They never get fruit. They're decent shade trees but they push up the sidewalk.
The yellow mimosa described by nathaliecraft is apparently an acacia. I looked up "acacia" in the plant files and found many kinds, and quite a few photos of the small round yellow flowers mentioned. This acacia, aka wattle, is quite common in Provence; it is evocative of the Mardi Gras carnival in Nice (as it is blooms in February) and grows well here in Corsica, too. A charactericstic it apparently shares with the pink mimosa treated in this article is its tendency to spread! My husband doesn't want any in our yard, despite its beauty. So I try to be satisfied with the absolutely gorgeous mimosas I see in our area.
The flowers smell very good, and a bouquet of them in the house brightens up the end of winter, even though the spherical flowers tend to close up into an even smaller sphere once cut from the tree. Florists sell varieties that last longer inside. The flowering period is about 6 weeks long.