Life will try to survive in even some of the harshest environments. Plants love sun, and the desert has plenty of sun. But plants normally need a lot of water to carry out various processes. Since water and other resources are scarce in the desert, desert plants have to develop a variety of common characteristics in order to survive.
Many desert plants have the ability to remain dormant during very dry periods, only to start growing when they finally have water. Therefore, many desert trees might appear dead when they are simply dormant. A tree that appears to be dead for a long period but revitalizes itself when it receives water is likely a desert tree, though some non-desert trees--like the resurrection fern--also have this characteristic.
Some plants, especially trees, develop very long roots that travel deep into the water table under the desert to absorb water. This is why some trees that appear to have no water source are able to survive in the desert. Other trees spread their roots out broadly so that they can absorb large amounts of water when there is rainfall. Trees with much shorter roots are likely not desert trees.
Many plants reduce the number of leaves and flowers that they have to cut back on the amount of water that is lost through transpiration. For example, bastard toadflaxes lose their leaves in the fall and the ponderosa pine has pine needles for leaves. Plants that have lots of broad leaves are likely not desert plants.
Like the leaves, the flowers of desert plants tend to be simplified. Also, the flowers blossom less often. The flowers of the bastard toadflax lack petals and instead have five septals. Toadflax blossoming occurs for two days and is open night and day. Plants in areas where there is more water can often afford to bloom for longer periods with larger flowers.
Most desert trees only reproduce when there is a heavy rainfall to ensure that their seeds are able to germinate and produce healthy offspring. Some desert tree seeds are able to remain dormant for a long time before germinating when there is sufficient water. Seeds that germinate easily likely did not come from a desert tree.
The farther into the desert that one travels, the more likely that the tree found is exclusively a desert tree. Some plants are more adaptive and can survive in wetter and drier locations. But plants living in the driest areas are definitely desert plants. Even some of the hardiest desert trees cannot survive deserts that get dry enough, such as with the foothill palo verde.