Application of a directed almost tree Is there anything interesting to say about an "almost BST" with directed edges?
Is there a domain where this structure naturally arises? I'm imagining a tree like the one below, but it would of course be more interesting if the tree were deeper with just a few of these strange edges.



I found this article on circuit rank which uses the term "almost tree", but it talks about circuits in undirected and directed graphs. What I'm describing is not a cycle.
What you are looking for are directed acyclic graphs (DAGs). If you want your DAG to have a unique "root", then you can ask for a DAG with a single source. DAGs are very useful in many branches of computer science.