Why is my desert Willow losing all of its leaves? The leaves have brown spots on them.
Over half of the branches are bare of leaves. the plant is definitely not thriving.
I would suspect that your heat/drought conditions and appropriate deep watering should be your first consideration.
Desert willow trees are know to be relatively resistant to pests and diseases, but there are some possibilities for disorders:
Nutrient and mineral excesses
Poor water management
It is not possible for us to give a definitive diagnosis without more information on growing conditons and management. You may need a site visit from a local consulting arborist or university extension service advisor.