We consider the case where the graph G is a tree. We construct a family of trees G on n vertices and pairs of search trees on G such that the minimum number of rotations required to transform one search tree into the other is Omega(n log n). This implies that the worst-case diameter of tree associahedra is Theta(n log n), which answers a question from Thibault Manneville and Vincent Pilaud. The proof relies on a notion of projection of a search tree which may be of independent interest.