In the photo above, I used rule of thirds where I placed the flower on the side of the frame. This photo shows balance making the viewer look at the photo more natural letting the eyes wonder.
In this photo I use the lines of the rain drain as a way of patterns and repetition.
The photo you selected uses the fence more or less where the imaginary grid of the rule of thirds would fall. However, the flower, the main subject, is still way too close to the center to consider this an example of the rule of thirds.