According to Serre's definition (in Serre's Trees):
($G$,$T$) is a tree of groups if T is a tree and there are groups $G_v$ and $G_e = G_\bar e $ for each $v\in vertT$ and $e \in edge T$, where $\bar e$ is the inverse edge of $e$ , together with monomorphsim $G_e \longrightarrow G_v$ if $v$ is an endpoint of $e$.
And $G_T=\varinjlim (G,T)$ is the direct limit of the system.
There is a property that the canonical maps from $G_v$ and $G_e$ to $G_T$ are injective. (Before Theorem 9 in page 38 of Serre's Trees)
I had proved the case of finite trees by useing example (b) in page 37 and induction.
But I have difficulties for proving the case of infinite trees.
For the case of infinite trees, all I had proved are in the following.
If there is a sequence of finite subtrees $T_n$ such that $T_0 \subset T_1 \subset ...\subset T $ with $T=\cup T_n$
and $vertT_{n+1} =vertT_n \cup$ {$p_{n+1}$} and $edgeT_{n+1}=edgeTn \cup ${$e_{n+1}, \bar e_{n+1}$},
where $v_{n+1}$ is an endpoint of $e_{n+1}$ not in $T_n$,
then $\varinjlim G_{T_{n}} \simeq G_{T}$.
And if the canonical maps $G_{T_n}$ to $\varinjlim G_{T_{n}}$ are injective, then the canonical maps from $G_e$ and $G_v$ to $G_T$ are injective.
I had no idea to choose a seqence of finite subtrees with the properties above even $vertT$ is countablely infinite and to prove the canonical maps are injective. Could someone help me?
There is a more general theory of graphs of groups $\cal G$, of which finite trees of groups are just a special case. The property you want generalizes to this context: associated to $\cal G$ is a kind of ``fundamental group'' $G_{\cal G}$, although this is not always any kind of direct limit; and all vertex groups and edge groups $G_v,G_e$ inject into $G_{\cal G}$.
While I cannot comment on how to alter the proof outline that you got stuck on, I'll suggest the alternative method of proof in the paper of Scott and Wall entitled "Topological methods in graph theory". As the title suggests, their proof is purely topological.