Suppose a group acts on a tree without edge inversions, and only the identity fixes a vertex. Then is a free group. Equivalently, the induced action on the geometric realization is free on every point: no nonidentity element fixes a vertex or an edge midpoint.
The quotient is a graph, and the map from the tree is a covering graph. Indeed, two incident edge germs with the same image would be related by a vertex stabilizer, which is trivial. Since the tree is simply connected, it is the universal cover of the quotient. Choosing lifts of a maximal subtree identifies with the quotient graph's fundamental group, freely generated by the edges outside that subtree.
The no-inversion hypothesis matters. The order-two group acting on a single edge by swapping its endpoints fixes no vertex, yet fixes the midpoint. That group is not a free group, and its quotient map folds the edge instead of giving a covering graph. Barycentric subdivision exposes the midpoint as a fixed vertex, removing this ambiguity.
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