There is a paper: Covering theory for graphs of groups, by Hyman Bass, Journal of Pure and Applied Algebra (1993) Volume: 89, Issue: 1-2, Pages: 3-47.
There is defined the notion of a covering of one graph of groups by another graph of groups. CoveringsAll such coverings of a given graph of groups are in correspondence with the subgroups of the fundamental group of the given graph of groups.