Updated
I found that I have misunderstood your question. I think you want to select the DOM parent element while you want to select the parent node in the tree view. This is my updated answer.
Midify your handler a bit:
function treeviewSelect(e) {
$('#relationshipsTree div').removeClass('relationshipsTree');
$(e.node).parents('li').first().children('div').addClass('relationshipsTree');
var node = this.dataItem(e.node);
window.open(node.NotificationLink, "_self");
}
A demo updated here