I know and understand the rule of thirds, but I would like to know, when taking a picture of something such as a tree, should the horizon be positioned on the top or bottom third line?
The most important thing to remember is that the rule of thirds is more like what you'd call "guidelines" than actual rules. Even if you're familiar with the rule, I suggest refreshing yourself on the background of the rule of thirds and where it comes from. It contains less ancient wisdom than many people assume.
So, since there's no hard and fast rule, that means there's really no magic answer to this. (Take a look at When is it OK to place the subject in the middle of a picture? and When should I break the rule of thirds?) It would be convenient if there were an always-correct answer, but if there were, all compositions would be identical and boring. We'd not even need photographers; robot cameras could be programmed to follow all the rules and produce great art. Clearly, that's not the case.
But that doesn't mean this is a bad question or that the rule of thirds is useless. It may be just a guideline, but it does have some basic good sense. My main advice is to not get too caught up on precise alignment with the ⅓ line or any other special divisions. If you keep the numerology out of it, though, there are some basic effects that differ when you place the horizon high or low.
A high horizon can emphasize the distance and three-dimensionality in your photo, if the landscape sweeps back towards it. Or, if the foreground is the main concern and there's a dominant foreground subject, putting the horizon high places the emphasis on that subject with the sky out of the way.
A low horizon can emphasize a majestic sky, and if the foreground is boring, you can just cut it out. This composition can have a more formal, observational feel, and a sense of looking into the picture rather than being in the picture. With low horizon, the subject is more apt to break the horizon line, which is a topic all unto itself.
A centered horizon tends to create a more static composition, and often when you choose this, it's because your photograph is about the reflection of the sky and the landscape. In this case, you would generally either want to emphasize that doubled relationship, or have another strong compositional element to create tension. The second image here provides a very successful example of this.
There are even more options. Horizon may be diagonal, vertical, it may be upside down. As Matt said, please do not consider any "rules" to be rules for yourself :)
One more thing: if the horizon is just slightly non-horizontal, it often grabs viewers' attention and "feels awkward". I'm not saying it's a bad thing but it may take viewer's attention off from the object/scenery that you want to be noticed.
If upper branches of a tree are more important, use the bottom line. If on shot you have nice shadow of this tree, use upper. This is dependant on your subject. Here my simple sample, if somebody don't know how tree looks ;) Here is "like a" rule of third sample, because pure rule is mostly boring.