(joint work with R. Grigorchuk ad D. Horadam) The tree representation theorem represents a certain group associated with the scale of an automorphism of a t.d.l.c. group as acting by symmetries of a regular (unrooted) tree. It shows that groups acting on regular trees are a fundamental part of the theory of t.d.l.c. groups.
There is also an extensive theory of self-similar and self-replicating groups of symmetries of rooted trees which has developed from the discovery (or creation) of examples such as the Grigorchuk groups.
It will be seen in this talk that these two branches of research are studying essentially the same groups.