What is this algorithm? Create a tree's equivalent hierarchical network
Your question contains the implicit assumption/premise that this transformation has a name.  I don't know why you would expect it to have a special name.  The transformation is not anything special or complicated and as far as I know it does not have any special name.  It appears that you are just taking a tree, and making a second copy of it (with all edges reversed; with all nodes cloned), and then just gluing the two copies together (identifying both copies of each leaf). 

For future reference, I don't think questions like "please tell me if this function has a special name" are a good fit for this site.  This site is for technical questions that admit an objectively evaluable answer and that relate to a real problem you have.  How do you expect to evaluate an answer that says "No, it doesn't have a special name"?  There is no reference one can cite to back up that answer, and no objectively verifiable evidence one can provide to support such an answer.  Moreover, questions about naming aren't a very good fit.  How will knowing whether there is a special name help  you solve a real problem?  (If you were thinking "it might help me find references that help me solve problem X", then you would be better off by just describing problem X.)