It seems my pretension has forced me to bite off more than I had planned.
No, the Tree of Life isn't a spread, it is a diagram illustrating an occult way of thinking called Qabalah, and the astrological attributions we are talking about derive mainly from the Hermetic Order of the Golden Dawn, a magickal order that worked to a great extent with Qabalah.
http://www.digital-brilliance.com/contributed/theatre/GDtarottol.jpg
Tarot wasn't originally part of Qabalah, but over the years occultists pinned the cards onto the Tree, a process which (arguably) reached its zenith with the Golden Dawn. As to the connection with the Devil and the World, it is almost impenetrably complicated, but I'll try to simplify, although I don't fully understand it myself.
Generally speaking, the Devil represents the good things in life (although some would say he represents taking them to extremes); food, drink, drugs, sex, the creative urges. The Tree represents in most part the astral plane, the planning stage of the world, and the World (or, in some decks the Universe) represents the final fruition of Creation, where we all live and breathe. One of the most important attributes of Saturn is Time and Death, a price we all pay for being alive. This isn't all bad, as together with the unpleasantness we gain enjoyment, and the Devil is on an area of the Tree representing the birth of consciousness, of Eve eating of the apple and gaining the urge to create, in essence, to become like the Creator. Capricorn ruled by Saturn exemplifies this, as it is all done on this plane, where babies are made. Thought of as more abstractly, the influence of the Devil on the World is why things seem "real" down here.
All the Tarot cards are connected to each other in some way, and thinking about them this expands knowledge of the cards and life in general. Talk to me if I didn't explain myself well (I probably didn't).
