A tree generated automatically
Trees are the structures created both by the automatic generation of the map, and by players. They are built of trunk blocks and leaves.
When generated by the map, they are frequently found in groups, in the style of a forest.
Trees come in 3 different heights and do not all grow exactly the same.
They can be generated in a third way as well. If a shrub is placed in sunlight on a grass block with enough space, a tree will likely grow within several seconds.
This feature was added to Creative Mode in version 0.28_01. But since this was removed due to its griefer-friendly applications ("treespam"), it was added to server.properties of the server files.
JTE: A command (/tree) allows you to build trees without a hassle. When you type the command, place a trunk and it'll automatically grow. Obviously this is just a custom server, so if you do it on another server, it will not work.
Indev Trees
A tree that was grown in a cave. Note the torches under the tree that gave the necessary light.
Currently, in Indev mode, trees wither and die if the leaves no-longer have a connection to a piece of soil below them and lost their connection to a tree trunk on the same height. The tree trunk can be cut from under the leaves to have a floating tree top. Low-lying trees will sometimes leave pillars of leaves that touch the ground and will not wither.
When planting trees in Indev mode they do not require grass to grow and must be planted at least 3 blocks away from other obstacles, they take around 1-2 minecraftian days in order to reach maturity. They can also be grown anywhere that sufficient light and space exist, including underground with torchlight.
After some research it was shown that when shrubs are planted near each trees other they usually grow bigger. This has not entirely been proven though. More input from other players would be appreciated.
Infdev Trees
Trees were updated yet again in Infdev, and now are significantly taller, with more realistic distribution of leaves.