[Turing_Machine]

To take a familiar example, the Web is not a tree. While you might argue that each page has unique children (outlinks), there is no unique parent -- a page can have an arbitrarily large number of incoming links, and the page can link back to those pages in turn, producing cyclic structures of immense complexity.

For some interesting discussion on the limits of tree structures with respect to urban planning see A City is Not a Tree by Christopher Alexander (who is also the inspiration for the Design Patterns movement in software engineering). http://www.rudi.net/node/317

[Deestan]

All good, just a nitpick: You don't need a unique parent to have a tree. You don't even need directed edges. :)

The purest definition of a tree graph is "a graph with no cycles".

[sesqu]

We use different definitions. My trees are acyclic connected graphs with one root.