A Note on the Equiseprable Tree

Let T be a tree and n(e T) denote the number of vertices of T, lying on the two sides of the edge e. Suppose T and T are two trees with equal number of vertices, e T and f T. The edges e and f are said to be equiseprable if either n(eT)=n(fT) or n(eT)=n(fT). If there is an one-to-one correspondence between the vertices of T and T such that the corresponding edges are equiseprable, then T and T are called equiseprable trees. Recently, Gutman, Arsic and Fortula investigated some equiseprable alkanes and obtained some useful rules (see J. Serb. Chem. Soc. (68)7 (2003), 549-555). In this paper, we use a combinatorial argument to find an equivalent definition for equiseprability and then prove some results about relation of equiseprability and isomorphism of trees. We also obtain an exact expression for the number of alkanes on n vertices which three of them have degree one.