who created the wizards staffs? or did they have to bring them with them when they arrived in middle earth?

I think the Istari are described as having the staves when they arrive in Middle-earth. They do seem to be connected to the power of the Wizards. Perhaps when the Istari were incarnated into physical bodies their staves were crafted to help them to channel and focus their power.

I wonder are they made of special timber from valinor from the time of the two trees that gives them special properties or something