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!