Near automorphisms of trees with small total relative displacements

For a permutation f of the vertex set V(G) of a connected graph G, let δ _f (x,y)=|d(x,y)−d(f(x),f(y))|. Define the displacement δ _f (G) of G with respect to f to be the sum of δ _f (x,y) over all unordered pairs {x,y} of distinct vertices of G. Let π(G) denote the smallest positive value of δ _f (G) among the n! permutations f of V(G). In this note, we determine all trees T with π(T)=2 or 4. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday.