Dojo example: http://dojo.telerik.com/ePEPAtEl/3
The DropDownTree's height is set to "auto" and there are enough nodes in the embedded treeview, so that the when the popup shows up a vertical scrollbar appears. In Chrome and IE, when the scrollbar appears the page is scrolled down automatically, which causes the popup to close.
height is not set explicitly in the DropDownTree (remove the setting from the dojo example). If the browser window is resized so that there isn't enough space for the whole popup to be displayed, on the popup showing up, the same behavior as in 1. is observed.
The popup closes automatically.
The popup remains open.