In the area of programming languages and compilers, we usually have trees without parent links, with substructure sharing, which are DAGs. We call them trees because most of the code operating on them thinks they are.
E.g. the Lisp expression (+ a (* b b)), informally called a syntax tree, is actually a DAG. Both apparent instances of the b symbol are actually graph edges leading to the same symbol object. Moreover, nothing points to its parent; that would be horrible.
Other kinds of trees don't always have parent pointers. For instance search trees like red-black can be implemented with parent pointers, or without; e.g. with functional recursion.
> E.g. the Lisp expression (+ a (* b b)), informally called a syntax tree, is actually a DAG
That is a syntax tree. Syntactically it is a tree. The fact that you have a later step that recognises that the `b` identifiers refer to the same object and turn it into a DAG doesn't mean it isn't a tree initially.
Guess the parent means that it's relatively straight forward to converted a DAG to a tree by duplicating nodes (or rather, sub-graphs) having multiple parents.
