Well, now that you 're asking...
I usually find myself wanting to store a tree in a database.
When i request a leaf by ID, i want to get the path, and when i get a leaf by path, i want to know its id.

Also, it's nice to know the previous/next (sibling) of a leaf.
I currently store the tree in a table like so:

id parent node
1  0      Root
2  1      Fruit
3  2      Apple
4  2      Pear
5  2      Banana
6  1      Vegetable
7  6      Spinach
[download]

I think you get the idea. The items usually aren't sorted like they are in the example above.