= Edge inversion
An edge inversion in a <group action on a tree> occurs when a group element preserves an edge as a set while exchanging its two endpoints. Thus setwise edge preservation is weaker than fixing the edge pointwise.
For the tree consisting of one edge with endpoints $u,v$, the order-two action swapping $u$ and $v$ is an inversion. It has no fixed vertex, although the midpoint is fixed in the geometric realization. Arguments requiring a fixed vertex must therefore exclude inversions or change the tree.
<Barycentric subdivision> inserts one new vertex at the midpoint of every edge. The induced action preserves the old and new vertex types. Each subdivided edge joins different types, so its endpoints can no longer be exchanged. This removes all edge inversions while retaining the original action on the underlying geometric tree.
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