Here is a slightly different take on this question (Find the tfs path of merged branch), different enough to deserve its own question.
In the above link, the authors describe how you can programmatically traverse the folder hierarchy using TFS API to determine the parent branch object from a folder or item within a branch, and then you can get the branch information for that branch.
However, a different question is: how do you use TFS API obtain a path to a different branch of the original folder or item, taking into account that the item may have been renamed or moved in the other branch?
For instance: if I have an object $/MyProject/Main/Foo/bar.txt, I can easily figure out that the parent branch is $/MyProject/Main using the method described. However, let's I also have the following versions of this file out there that were created by branching the original files:
$/MyProject/Dev/Foo/bar.txt
$/MyProject/Experiment/Foo/bar.renamed
$/MyOtherProject/Main/openbar.woohoo
How can I determine the list of references to the specific files or folders in a branch? Clearly there is a way to do this because Visual Studio's merge dialog automatically populates the list of available targets with all references to the source file or folder, even if the source is not a branch.