In order to maintain the structure of the tango tree (auxiliary trees correspond to preferred paths), we must do some updating work whenever preferred children change as a result of searches. When a preferred child changes, the top part of a preferred path becomes detached from the bottom part (which becomes its own preferred path) and reattached to another preferred path (which becomes the new bottom part). In order to do this efficiently, we'll define "cut" and "join" operations on our auxiliary trees.