1. At first I have taken a picture of the leaf on the plain white snow with no other objects on the background. As the leaf is the only object on the picture, I have placed it on the left side, but vertically in the middle of the picture. I that grains of snow from a bit blurred white to clearly focused are driving the attention to the center of the leaf. Though, it might make the composition a bit static. I tried several other options. Placing the object vertically closer to the edges, leaves too much white background. Placing the point centrally, makes composition really static.
2. For the second shot I have chosen an object with a little bit different background to see, which composition will work better in this case. The trees on the background are clearly dividing the frame. And I decided to place the point horizontally on the right side and vertically in the center of the remaining frame. I am not quite sure that this composition works, it might be worth putting the point vertically a bit closer to the edge...
3. For the third part of of the experiment I have chosen the background other than snow. The office building with some lights and a lantern on this background seemed to work well. I have placed the object exactly on the line with windows. The light from the building slightly moves attention to the lantern. The small reflections in the bottom part of the picture seem to complement the composition.
Summarizing the exercise, I would say that though there are no clear rules for placing the point in one or another part of the frame, the central position is very unlikely to be justified. At least, none of my objects worked centrally. Generally, placing the point within the frame is highly driven by the type of the background.