I need to find references to the following two facts (if a second one has a short proof then I can use it instead)
1) Each automorphism of oriented tree has a fixed point or invariant line.
2) Let group $G$ be a free product of finitely many finite groups $G_i$: $G=G_1\ast \cdots\ast G_n.$
If elements $g_1=a$ and $g_2=a^b = b^{-1}ab$ have an infinite order in $G$ and $[g_1,g_2]\ne 1$ in $G$, then $\left \langle g_1^2,g_2^2 \right \rangle\simeq F_2$, where $F_2$ is a free group of rank two.
1.) Each automorphism of a tree fixes a vertex or an edge.
Proof: Let $T$ be a tree. If $|V (T )| = 1$ then clearly the identity automorphism must fix the vertex. If $|V (G)| = 2$, there are two automorphisms - the identity and the map that switches the two vertices. In the latter case, the edge is fixed. Now suppose for all $T$ with $|V (T )| ≤ n$, every automorphism fixes either an edge or a vertex. Let $|V (T )| = n + 1$ and let $\{v_1, \ldots , v_k\} ∈ V (T )$ be the set of vertices with degree one. We note that this set is nonempty as $T$ is a tree. Let $\phi$ be an automorphism of $T$ . Then, $\phi(v_i) ∈ \{v_1, \ldots , v_k\}$ for $1 ≤ i ≤ k$ as automorphisms preserve degrees. Consider $T′ = T \setminus \{v_1, \ldots , v_k\}$. Then $\phi$ is also an automorphism of $T′$, and $T′$ is also a tree, and $|V (T')| < n+1$. By the induction hypothesis, $\phi$ fixes an edge or a vertex of $T′$. Thus, $\phi$ fixes an edge or a vertex of $T$.
