We explore the group of 2x2 determinant 1 matrices with coefficients inthe ring of integers Z as well as an analog obtained by replacing Z by thering of polynomials F[t] where F is a finite field. Decomposition theoremsfor these groups will be obtained using the actions of the groups ontrees, which are connected graphs having no cycles, such as that of degree3 a part of which is pictured above.REFRESHMENTS WILL BE SERVED!