  On trees of fixed order, we show a direct relation between Kemeny's constant
and Wiener index, and provide a new formula of Kemeny's constant from the
relation with a combinatorial interpretation. Moreover, the relation simplifies
proofs of several known results for extremal trees in terms of Kemeny's
constant for random walks on trees. Finally, we provide various families of
co-Kemeny's mates, which are two non-isomorphic connected graphs with the same
Kemeny's constant, and we also give a necessary condition for a tree to attain
maximum Kemeny's constant for trees with fixed diameter.
