by permuting the list of little disks. The operadic composition for little disks is illustrated in the accompanying figure to the right, where an element is composed with an element to yield the element obtained by shrinking the configuration of and inserting it into the i-th disk of , for . Analogously, one can define the little n-disks operad by considering configurations of disjoint n-balls inside the unit ball of . Originally the little n-cubes operad or the little intervals operad (initially called little n-cubes PROPs) was defined by Michael Boardman and Rainer Vogt in a similar way, in terms of configurations of disjoint axis-aligned n-dimensional hypercubes (n-dimensional intervals) inside the unit hypercube. Later it was generalized by May to the little convex bodies operad, and "little disks" is a case of "folklore" derived from the "little convex bodies". Rooted trees In graph theory, rooted trees form a natural operad. Here, is the set of all rooted trees with n leaves, where the leaves are numbered from 1 to n. The group operates on this set by permuting the leaf labels. Operadic composition is given by replacing the i-th leaf of by the root of the i-th tree , for , thus attaching the n trees to and forming a larger tree, whose root is taken to be the same as the root of and whose leaves are numbered in order. Swiss-cheese operad The Swiss-cheese operad is a two-colored topological operad defined in terms of configurations of disjoint n-dimensional disks inside a unit n-semidisk and n-dimensional semidisks, centered at the base of the unit semidisk and sitting inside of it. The operadic composition comes from gluing configurations of "little" disks inside the unit disk into the "little" disks in another unit semidisk and configurations of "little" disks and semidisks inside the unit semidisk into the other unit semidisk. The Swiss-cheese operad was defined by Alexander A. Voronov. It was used by Maxim Kontsevich to formulate a Swiss-cheese version of Deligne's conjecture on Hochschild cohomology. Kontsevich's conjecture was proven partly by Po Hu, Igor Kriz, and Alexander A. Voronov and then fully by Justin Thomas. Associative operad Another class of examples of operads are those capturing the structures of algebraic structures, such as associative algebras, commutative algebras and Lie algebras. Each of these can be exhibited as a finitely presented operad, in each of these three generated by binary operations. For