What branch has a appointed trustee of state university?

The branch of government that appoints a trustee of a state university varies by state. There's no single answer. It's often the executive branch, through the governor or a board appointed by the governor. However, in some states, the legislative branch (the state legislature) may have a role in the appointment process, either directly appointing trustees or confirming gubernatorial appointments. Some states may also have a combination of executive and legislative involvement.