The title of this question pretty much says it all.
I would like to know if there is some standard name for trees in which, if any two nodes are siblings, then either they are both leaves, or they are both non-leaves.
One variation of the same situation occurs in trees with two types of nodes, such that one of these two types can have children, and the other one cannot. (An everyday example of this is a simple file system consisting exclusively of "folders" and "files".) In this case, the condition in the title could be modified to requiring that siblinghood can happen only between nodes of the same type. (With this variation, the number of children a node happens to have plays no role in whether the tree containing tree satisfies the condition or not.)
Is there a standard name for either of the two variations described above?
(P.S. Should I post this in math.stackexchange.com instead.)