The phrase “tributary system” originated in the 1940s and was originally intended to describe a set of varied practices, not a simple, hierarchical system. You and your students will find it helpful to remember that Ming China had a toolbox of diverse foreign policy options. These included investiture of specific leaders, engaging in different kinds of diplomatic communication, sending and accepting diplomatic embassies (including tribute missions), strategic marriages, allowing/disallowing entry into Chinese territory, trading/refusing to trade, payments to troublesome foreign leaders (in silk, for example), stationing troops on land or patrolling sea lanes, going to war/sending troops to aid in conflict, and more. They varied their approach across time and place, depending on circumstances. Other polities also had foreign policy options for interacting with each other and with China. Yes, China was powerful, but East Asian polities did not all agree that they were part of a single, simple hierarchy with China at the top.