  In this paper we prove that two quantities relating to the length of
permutations defined on trees are independent of the structures of trees. We
also find that these results are closely related to the results obtained by
Graham and Pollak (Bell System Tech. J. 50(1971) 2495--2519) and by Bapat,
Kirkland, and Neumann (Linear Alg. Appl. 401(2005) 193--209).
