Fruit bearing Olive trees don't fair too well here in the desert Southwest, the winter temps get too low and the tree freezes back to the ground, but it does seem to come back from the roots in the spring.
There are or shall I say were some fairly large fruitless Olive trees in Las Cruces, NM, but with the recent below zero temps, they died and are coming back as shrubs.
It is very hard to find fruit bearing trees in any of the local nurseries, because of the pollen they produce and related allergy issues. Arizona has a lot of Olive trees, but they are non-bearing.
If you really want a bearing Olive tree and have a well protected place to plant it, buying one on-line is your best choice. There is a link on the left for Olive tree seeds.
The top three photos to the right were taken in Greece. They have mountain-sides filled with Olive trees, very beautiful!
There is a variety of Olive that will grow here and it is the NM Olive or NM Privet. They are more like a shrub with very small berry like fruit.
They do very well here in the desert Southwest and can even be found growing wild in some arroyos. Birds love the berries it produces.