Patching File System Trees with PowerShell

While working on a elaborate PowerShell script I needed to find a solution for the following problem: I wanted to do a string replace operation on a file system path name and have the resulting new string be reflected by the file system. For example the path name "C:\dogs\dog food\my favorites\hot dogs\Who let the dogs out.mp3" should be renamed by replacing "dog" with "cat". The resulting path name would be "C:\cats\cat food\my favorites\hot cats\Who let the cats out.mp3". Sounds easy, but how do I move the folders and files that are referred to in the path to their new locations? And even better, how can I do this recursively? This short blog post demonstrates two possible approaches.