What's the proper name of that symbol to collapse/expand nodes in a directory tree?

Solution 1:

The Microsoft UX Guide mentions these UI elements in the section about Progressive Disclosure Controls.

The element shown in your screenshot is amazingly named rotating triangle. Their counterparts are named plus and minus controls:

Both elements are also discussed in more detail on later pages, but the name is always the same.

The name of another UI element that is often associated with collapsible containers is the chevron, which is also pictured in the first screenshot. But this element is used in a different context.

Solution 2:

Generic term is "disclosure widget". Third parties will of course use their own variations though.

https://en.wikipedia.org/wiki/Disclosure_widget

A disclosure widget, expander, or disclosure triangle is a graphical control element that is used to show or hide a collection of "child" widgets in a specific area of the interface. The widget hides non-essential settings or information and thus makes the dialog less cluttered.

The disclosure widget may be expanded or collapsed by the user; when this occurs, the containing window may be expanded to accommodate the increased space requirement. The state of the widget is often signified by a label with a triangle next to it, pointing sideways when it is collapsed and downward when it is expanded (corresponding to the widget's current state), or a button with an arrow pointing downward when it is collapsed and upward when it is expanded (corresponding to how the widget will change state if the button is clicked). Some disclosure widgets can appear as a plus button when collapsed and a minus button when expanded.

In some implementations, the widget may be able to remember its state between invocations; this may increase user familiarity with the interface. In other implementations, the widget may disappear when clicked in order to make room for the newly revealed controls; this state is not remembered.

Solution 3:

Technically spoken, this symbol is a button to collapse/expand the TreeViewControl. Microsoft not provided any collapsible control for Windows applications, if we required Collapsible control we have to develop our own control with our own functionality.

So I would say the most generic name of this control is "Collapsible Control" :)