Basically you are correct here in your description of the hammock, but the tree would also have some bending - see the discussion below on bending. Regarding the diving board, engineers would refer to the diving board as a bending member, as opposed to a tension or compression member. You are correct in that bending members will have compression stresses on one side and tension stresses on the other side, depending on which way they are bent. We try to design domed structures so that all the stresses are in compression - that makes it an efficient structure. However, there are always uneven loads, such as wind loads, that must be resisted partly by bending in the dome which might cause some tension. So it is usually not all compression. The air pressure supported domes, such as we have in Vancouver, have a roof that is all in tension, but the ring at the bottom of the roof where the cables are attached is in compression. For your swing example, if the vertical supports are single posts anchored in the ground, they would be in compression if the person is sitting on the swing without moving. Once the person is swinging, there are horizontal forces as well, and these would cause bending in the supports. The horizontal beam is a bending member and would have tension on the side facing the swinger, and compression on the opposite side.
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