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Choose free group generators $a_1,\dots ,a_n$. Construct the Schreier coset graph $\Gamma$ with vertices the right cosets $Hg$ and a positively oriented edge labelled $a_k$ from $Hg$ to $Hg{a}_k$. This is well defined and connected. Its inverse edge carries $a_k^{-1}$. A word in the group generators determines a unique path from the base vertex $H$; that path closes exactly when the word represents an element of $H$.

Deleting a consecutive edge followed by its inverse corresponds exactly to free cancellation in the word. Since reduced words are unique in a free group, the resulting map from based loops modulo these cancellations to $H$ is injective as well as surjective, and respects concatenation. Thus $H$ is the fundamental group of $\Gamma$. More explicitly, choose a spanning tree $T$ and, for each unoriented edge outside $T$, choose one orientation. The tree path from the base vertex to that edge, followed by the edge and the returning tree path, defines an element of $H$. Every loop is a product of these elements and their inverses: insert the appropriate tree paths between successive edges, and tree edges contribute trivial factors. Collapsing the tree leaves one loop for each remaining edge. A product of the chosen basis loops maps to the corresponding word in these loop labels. If the original path could cancel to the constant path, its image could do so as well, since collapsing tree edges preserves every backtracking cancellation. A nonempty reduced word in the remaining labels cannot cancel to the empty word, proving that there are no further relations. These elements therefore form a free basis. This proves that $H$ is a free group without assuming the subgroup theorem.

If $i=[F_n:H]$ is finite, the graph has $i$ vertices and $ni$ unoriented edges, counted by their positive labels; loops and parallel edges count separately. A spanning tree has $i-1$ edges. The Nielsen–Schreier formula follows:For a sequence of finite indices $i_j\to \infty$, this gives the rank-to-index limit for subgroups of a free group

Finally, let $\theta :F_n\to F_n$ be an endomorphism, and put $L=\theta (F_n)$. It is generated by the $n$ images $\theta (a_k)$. If its index $d$ is finite, it is free of rank $r=1+d(n-1)$. A rank-$r$ free group needs at least $r$ group generators: its abelianization is ${\mathbb{Z}}^r$, and reduction modulo $2$ gives the $r$-dimensional vector space $(\mathbb{Z}/2\mathbb{Z}{)}^r$, which cannot be generated by fewer than $r$ vectors. ThereforeSince $n\ge 2$, this forces $d\le 1$, hence $L=F_n$. This proves that a finite-index image of a free-group endomorphism is surjective:

Let $r=d(G)$ be the rank of a group, let $S$ be a generating set of size $r$, and suppose $[G:H]=d$. Choose a Schreier transversal adapted to a spanning tree in the Schreier coset graph. By Schreier's lemma, $H$ is generated by the elementsThere are $dr$ candidates. The $d-1$ oriented edges in the spanning tree give trivial candidates, leaving at most $dr-(d-1)$. This proves the Schreier index-rank inequality

The based Schreier coset graph has vertices $⟨a⟩g$ and an $s$-edge from $⟨a⟩g$ to $⟨a⟩gs$. At the base vertex $⟨a⟩$ there is an $a$-loop. There are no other cycles: the fundamental group of the covering is the subgroup $⟨a⟩\cong \mathbb{Z}$, and its displayed loop already generates it.

Equivalently, start with one $a$-circle at the base vertex and attach labelled trees so that every vertex has exactly one incoming and one outgoing $a$-edge and exactly one incoming and one outgoing $b$-edge. After deleting the base $a$-loop, the underlying graph is a tree. This describes the complete infinite covering and distinguishes it from its finite core, which is just the $a$-loop.

Use the Schreier coset graph of the subgroup, with one directed edge labelled $s$ from $Hg$ to $Hgs$ for each $s\in \{a,b\}$. The homomorphismidentifies the cosets of $\ker \varphi$ with the integers. The covering graph therefore has vertices $v_n$, $n\in \mathbb{Z}$, an $a$-edgeand a $b$-loop at every $v_n$. Thus it is a doubly infinite $a$-line with one $b$-circle attached at each integer vertex.