Counting and Averaging Problems in Graph Theory

Paul Gunther (1966), proved the following result: Given a continuous function f on a compact surface M of constant curvature -1 and its periodic lift g to the universal covering, the hyperbolic plane, then the averages of the lift g over increasing spheres converge to the average of the function f over the surface M. Heinz Huber (1956) considered the following problem on the hyperbolic plane H: Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x in H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. In this thesis, we use a well-known analogy between the hyperbolic plane and the regular tree to solve the above problems, and some related ones, on a tree. We deal mainly with regular trees, however some results incorporate more general graphs.