Check the underline note on page 603, which begins with "It may seem arbitrary that breadth-first search is limited to only one source whereas depth-first search may search from multiple sources ..."
The number of sources is the reason one is a tree (single root/source) and the other is a forest (multiple roots/sources).
PS. This is, of course, not some conceptual difference, but merely a choice made by the authors, for reasons explained in the underline note.