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We introduce the notions of tree-like path and tree-like equivalence betweenpaths and prove that the latter is an equivalence relation for paths of finitelength. We show that the equivalence classes form a group with some similarityto a free group, and that in each class there is one special tree reduced path.The set of these paths is the Reduced Path Group. It is a continuous analogueto the group of reduced words. The signature of the path is a power serieswhose coefficients are definite iterated integrals of the path. We identify thepaths with trivial signature as the tree-like paths, and prove that two pathsare in tree-like equivalence if and only if they have the same signature. Inthis way, we extend Chen's theorems on the uniqueness of the sequence ofiterated integrals associated with a piecewise regular path to finite lengthpaths and identify the appropriate extended meaning for reparameterisation inthe general setting. It is suggestive to think of this result as anon-commutative analogue of the result that integrable functions on the circleare determined, up to Lebesgue null sets, by their Fourier coefficients. As asecond theme we give quantitative versions of Chen's theorem in the case oflattice paths and paths with continuous derivative, and as a corollary deriveresults on the triviality of exponential products in the tensor algebra.Comment: 52 pages - considerably extended and revised version of the previous  version of the pape

Uniqueness for the signature of a path of bounded variation and the reduced path group