In information theory, the definition of the amount of self-information and information entropy is often expressed with the binary logarithm, corresponding to making the bit the fundamental unit of information. With these units, the Shannon–Hartley theorem expresses the information capacity of a channel as the binary logarithm of its signal-to-noise ratio, plus one. However, the natural logarithm and the nat are also used in alternative notations for these definitions. Combinatorics Although the natural logarithm is more important than the binary logarithm in many areas of pure mathematics such as number theory and mathematical analysis, the binary logarithm has several applications in combinatorics: Every binary tree with leaves has height at least , with equality when is a power of two and the tree is a complete binary tree. Relatedly, the Strahler number of a river system with tributary streams is at most . Every family of sets with different sets has at least elements in its union, with equality when the family is a power set. Every partial cube with vertices has isometric dimension at least , and has at most edges, with equality when the partial cube is a hypercube graph. According to Ramsey's theorem, every -vertex undirected graph has either a clique or an independent set of size logarithmic in . The precise size that can be guaranteed is not known, but the best bounds known on its size involve binary logarithms. In particular, all graphs have a clique or independent set of size at least and almost all graphs do not have a clique or independent set of size larger than . From a mathematical analysis of the Gilbert–Shannon–Reeds model of random shuffles, one can show that the number of times one needs to shuffle an -card deck of cards, using riffle shuffles, to get a distribution on permutations that is close to uniformly random, is approximately . This calculation forms the basis for a recommendation that 52-card decks should be shuffled seven times. Computational complexity The binary logarithm also frequently appears in the analysis of algorithms, not only because of the frequent use of binary number arithmetic in algorithms, but also because binary logarithms occur in the analysis of algorithms based on two-way branching. If a problem initially has choices for its solution, and each iteration of the algorithm reduces the number of choices by a factor of two, then the number of iterations needed to