The sandpile group of a bilateral regular tree

Let Tn be a (d − 1)-ary tree of height n. A bilateral regular tree is the multigraph T d n,n that is defined by : (i) juxtaposing two Tn’s, (ii) drawing a new edge between two roots and (iii) drawing d− 1 edges between each leaf and a new vertex, called the sink. In this paper the sandpile group of T d n,n is determined.