I solved the problem:
I searched through the net, and found this post:
http://markmail.org/message/zozobs24v4wyvf6p
and wondered if that quirk was causing my problem.
I already was using a subclass of Tree,
so added to the constructor:
DOM.setStyleAttribute(getElement(), "position", "static");
and; now the second tree is not disappearing when I select the first
tree.
I'm planning to file a bug on this issue.