The Shulhan Aruch (Orah Haim 336) codifies the prohibition against making use a tree such as climbing or leaning against it on Shabbat. The Sages enacted this prohibition out of the concern that one might remove fruits or branches, in violation of Shabbat. This prohibition includes making use of the side of a tree, such as by climbing on a ladder that leans against a tree. Even though one is not climbing the tree itself and will not even be touching the tree, it is nevertheless forbidden to climb a ladder leaning against a tree, since this involves direct use of the tree.
It is permissible, however, to use an object hanging from a peg attached to a tree. This situation is called "Sedadin Disdadin," or "sides of the sides," and it is permissible to use the tree in this fashion because such use is indirect. Thus, for example, if a peg is thrust into a tree, and a basket hangs from the peg, it would be permissible to store things in the basket on Shabbat, as this involves only indirect use of the tree.
These rules affect the question of whether on Shabbat one may lie in a hammock tied to a tree. Often, hammocks are suspended by two trees, with one end tied to one tree and the other end to a second tree. The permissibility of using the hammock in such a case depends on how precisely it is tied. If the hammock is tied directly to the tree, then it may not be used on Shabbat. Normally, however, hammocks are tied to poles that extend from the trees, and not to the trees themselves, and in such a case, the hammock may be used on Shabbat. This applies to swings, as well. If a swing hangs from a chain tied to a tree, the swing may be used, since it is not tied directly to the tree.
Summary: One may use on Shabbat a hammock that is tied to a tree, as long as it is not tied directly to the tree, but rather to a pole extending from the tree. Likewise, although one may not use on Shabbat a swing tied directly to a tree, one may use a swing hanging from a chain that is tied to a tree.
