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.
Application of a directed almost tree
graph theory;terminology;binary trees
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.