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We develop the theory of ``branch algebras'', which are infinite-dimensionalassociative algebras that are isomorphic, up to taking subrings of finitecodimension, to a matrix ring over themselves. The main examples come fromgroups acting on trees.  In particular, for every field k we construct a k-algebra K which (1) isfinitely generated and infinite-dimensional, but has only finite-dimensionalquotients;  (2) has a subalgebra of finite codimension, isomorphic to $M_2(K)$;  (3) is prime;  (4) has quadratic growth, and therefore Gelfand-Kirillov dimension 2;  (5) is recursively presented;  (6) satisfies no identity;  (7) contains a transcendental, invertible element;  (8) is semiprimitive if k has characteristic $\neq2$;  (9) is graded if k has characteristic 2;  (10) is primitive if k is a non-algebraic extension of GF(2);  (11) is graded nil and Jacobson radical if k is an algebraic extension ofGF(2).Comment: 35 pages; small changes wrt previous versio

Branch rings, thinned rings, tree enveloping rings