Chip-Firing on Infinite $k$-ary Trees

We use an infinite k-ary tree with a self-loop at the root as our underlying graph.We consider a chip-firing process starting with N chips at the root.We describe the stable configurations.We calculate the number of fires for each vertex and the total number of fires.We study a sequence of the number of root fires for a given k as a function of N and study its properties.We do the same for the total number of fires.