Is a church allowed to have Board of Trustee meetings in private, and not allow other members of the church to attend.

There are many reasons, discuss ,private employee issues, some legal issues,
So what does there incorp articles say,
What is meeting for

Certainly. The board of directors of a corporation (and this board of trustees is probably their name for the board of directors of a non-profit corporation) often meets without the entire membership in attendance. You would have to consult the portions of your state corporations code pertaining to non-profit corporations to see exactly what the board can and can not do.