Since a node has no direct control over the root, how can I change the root pointer when I have to?  (A one-child or no-child delete of the root node.)

Every node contains a pointer:  Tree *t.  This pointer points to the Tree object containing that node.  If a tree contains 1000 nodes, then every one of those 1000 nodes has a t pointer pointing to the Tree object.

From any method in the node class, I can say, for example,

t->setroot (nullptr);

This will call t (the Tree's setroot) method.

Or I can say t->setroot (anything else).

Every node in the tree contains four pointers:

One to the left child, one to the right child, one back to the parent, and one to the Tree object itself.