Amputation in photographic terms relates to the artful act of cutting off of limbs and other parts of the body in portraiture and group photography. (this could also include silhouettes). It applies not only people, but animals, pets, landscapes, trees, etc., they can all fall victim to this terrible tragedy.
Sometimes it is unavoidable and even necessary to amputate, but it is advisable to know the acceptable way of doing it, so as not to cause it to look out of place or too obvious.
A classic example of amputation is the traditional old "head-and-shoulders" portrait shot. Where the upper torso and the rest of the body have been severed, in the correct fashion, in order to produce that sort of photograph.
So, if you need to amputate, to make it look right, it is more acceptable to do it about half way between joints, like wrist and elbow, ankle and knee, and in the case of the head and shoulders shot, between the shoulders and the abdomen.
If the photograph is of an animal, or family pet, it is obviously better if you can get the entire animal into the scene, or use the same concept.
Incidentally, if when photographing an animal, (or anything else for that matter) side-on, give it plenty of space in front. It tends to balance things and the subject will look more aesthetically comfortable if it can see into the scene instead of out of it. Anyway, where were we? Ah yes, trees can also be a problem where amputation is concerned, especially on the edges of landscape shots. Items like these are known as "intrusions" - little distractions that partially poke into the fringes of the frame. It is better to have a whole tree than only half a tree. If you can't fit it all in, keep it out altogether. If you have to include it, try to include at least most of it. It certainly looks more acceptable to have more of it than just a bit of it.
Any intrusions that can't be helped at point of shoot could possibly be cloned out with your photo editing program. Remember, one of the age old rules of photography - keep it simple..."less is more!".
See also "Cropping"