What is the prognosis of an astrocytoma in a child?

Depends on tumor. There are a few different types of astrocytoma- some without much risk of recurrence and some which are aggressive. Sorry it's not more specific. If you know the tumor type, i can be more specific in my answer.
Many types. As stated, there are many types of childhood astrocytoma with many different treatment regimens. For a good summation for patients and their familes, see the national cancer institute's page at: http://www.Cancer.Gov/cancertopics/pdq/treatment/child-astrocytomas/patient/page1/allpages.