The rule of thirds applies in black and white photos as well. One key to black and white is that a photo should have nearly black and nearly white areas in any photo. This is the same as saying the photo has a long contrast scale. I believe this is a good idea for color photos as well, if possible.

Note how this tree twists and turns all directions. I placed it on the right of the frame with the trunk line along the left third line. The twists and turns are placed on the intersection points to the extent possible. To make the compositional placements, you must see the curves and twists in your subject. You will get better at this over time and it will become instinctive.

This ancient cedar bonsai is atop Goat Bluff in the Buffalo National River Park, a trail near the Cecil Cove Trailhead.