Let f be a polynomial in one variable with integer coefficients, and p a prime. A solution of the congruence f(x) 0(mod p) may branch out into several solutions module p(2), or it may be extended to just one solution, or it may not extend to any solution. Again, a solution module p(2) may or may not be extendable to solutions module p(3), etc. In this way one obtains the ''solution tree'' T = T(f) of congruences module p(lambda) for lambda = 1, 2,.... We will deal with the following questions: What is the structure of such solution trees? How many ''isomorphism classes'' are there of trees T(f) when f ranges through polynomials of bounded degree and height? We will also give bounds for the number of solutions of congruences f(x) = 0(mod p(lambda)) in terms of p, lambda and the degree of f.
