  Generalized associahedra were introduced by S. Fomin and A. Zelevinsky in
connection to finite type cluster algebras. Following recent work of L. Pournin
in types $A$ and $B$, this paper focuses on geodesic properties of generalized
associahedra. We prove that the graph diameter of the $n$-dimensional
associahedron of type $D$ is precisely $2n-2$ for all $n$ greater than $1$.
Furthermore, we show that all type $BCD$ associahedra have the non-leaving-face
property, that is, any geodesic connecting two vertices in the graph of the
polytope stays in the minimal face containing both. This property was already
proven by D. Sleator, R. Tarjan and W. Thurston for associahedra of type $A$.
In contrast, we present relevant examples related to the associahedron that do
not always satisfy this property.

Combinatorics