For topological space $X$ (connected, path connected etc.), there is classification of coverings of $X$ : for fixed $x\_0\in X$, consider $\pi\_1(X,x\_0)$. Then there is a $1-1$ correspondance between conjugacy class of subgroups of $\pi\_1(X,x\_0)$ and covering spaces of $X$ (upto isomorphism).
We define universal cover of $X$ to be a cover $\tilde{X}$ which is a cover of every cover $Y$ of $X$. This exist and unique up to isomorphism.


Now consider a *graph of groups* $(X,A)$, where $X$ is a graph and $A=(A\_v,A\_e)$ is a family of groups attached to vertices $v\in V(X)$ and edges $e\in E(X)$ of $X$ with injections from edge groups to end-vertex groups.


In the book *"Trees"-Serre*, the universal cover of $(X,A)$ is defined to be a connected graph $\tilde{X}$ with:


1) a morphism $p\colon \tilde{X}\rightarrow X$ of graphs;


2) an action of $\pi\_1(X,x\_0)$, $(x\_0\in V(X))$ on $\tilde{X}$ such that stabilizer of $\tilde{v}\in p^{-1}(v)$ is isomorphic to vertex group $A\_v$, $v\in V(X)$


(In other words, it is a graph, with action of $\pi\_1(X,x\_0)$, $(x\_0\in V(X)$, such the quotient graph of groups $X/\pi\_1(X,x\_0)$ is isomorphic to given graph of groups).


***Question***: Is there a construction of universal cover of a graph of groups analogous to the construction of universal cover of topological spaces ( i.e. a cover of every cover of given graph of groups).


As an illustration, how can we obtain all coverings of $(X,A)$ where $X$ is the graph $\circ --\circ$ with vertex groups $\mathbb{Z}/l$, and $\mathbb{Z}/n$ and edge group trivial.



Coverings of a graph of groups
The universal cover of a graph of groups *is* the Bass--Serre tree. This is described in Serre's book *Trees*, to which you refer. I don't have a copy to hand, so I can't give you the precise reference, but it's the main object of study throughout the book.


Let me also add that, when trying to think about coverings of graphs of groups, I find it much easier to think about coverings of graphs of spaces. The initial reference for this is paper of [Scott and Wall](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=scott&s5=wall&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=564422). My students wrote some things about this point of view in [this blog](http://392c.wordpress.com/).


That said, it's not always feasible to describe every finite-sheeted covering of a graph of groups. For instance, in the case that you ask for, the set of coverings that you ask for is in bijection with the set of transitive permutation groups generated by a pair of permutations of order $l$ and $m$. Just the set of *regular* coverings is in bijection with the set of groups $\Gamma$ equipped with a generating set $\{a,b\}$, where $a$ is of order $l$ and $b$ is of order $m$. This is an unimaginably complicated set!


