  An elimination tree for a connected graph $G$ is a rooted tree on the
vertices of $G$ obtained by choosing a root $x$ and recursing on the connected
components of $G-x$ to produce the subtrees of $x$. Elimination trees appear in
many guises in computer science and discrete mathematics, and they encode many
interesting combinatorial objects, such as bitstrings, permutations and binary
trees. We apply the recent Hartung-Hoang-M\"utze-Williams combinatorial
generation framework to elimination trees, and prove that all elimination trees
for a chordal graph $G$ can be generated by tree rotations using a simple
greedy algorithm. This yields a short proof for the existence of Hamilton paths
on graph associahedra of chordal graphs. Graph associahedra are a general class
of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov,
whose vertices correspond to elimination trees and whose edges correspond to
tree rotations. As special cases of our results, we recover several classical
Gray codes for bitstrings, permutations and binary trees, and we obtain a new
Gray code for partial permutations. Our algorithm for generating all
elimination trees for a chordal graph $G$ can be implemented in time
$\mathcal{O}(\sigma)$ on average per generated elimination tree, where
$\sigma=\sigma(G)$ denotes the maximum number of edges of an induced star in
$G$. If $G$ is a tree, we improve this to a loopless algorithm running in time
$\mathcal{O}(1)$ per generated elimination tree. We also prove that our
algorithm produces a Hamilton cycle on the graph associahedron of $G$, rather
than just Hamilton path, if the graph $G$ is chordal and 2-connected. Moreover,
our algorithm characterizes chordality, i.e., it computes a Hamilton path on
the graph associahedron of $G$ if and only if $G$ is chordal.
