  In a recent paper [3], the authors introduced a map $\mathcal{F}$ which
associates a Deitmar scheme (which is defined over the field with one element,
denoted by $\mathbb{F}_1$) with any given graph $\Gamma$. By base extension, a
scheme $\mathcal{X}_k = \mathcal{F}(\Gamma) \otimes_{\mathbb{F}_1} k$ over any
field $k$ arises. In the present paper, we will show that all these mappings
are functors, and we will use this fact to study automorphism groups of the
schemes $\mathcal{X}_k$. Several automorphism groups are considered:
combinatorial, topological, and scheme-theoretic groups, and also groups
induced by automorphisms of the ambient projective space. When $\Gamma$ is a
finite tree, we will give a precise description of the combinatorial and
projective groups, amongst other results.
