A NOTE ON THE EQUISEPARABLE TREES

Let T be a tree and n_{l}(eIT) and n_{2}(eIT) denote the number of vertices of T, lying on the two sides of the edge e. Suppose T_{l} and T_{2} are two trees with equal number of vertices, e in T_{1} and f in T_{2}. The edges e and f are said to be equiseparable if either n_{l}(eIT_{I}) = n_{l}(fIT_{2}) or n_{l}(eIT_{I}) = n_{2}(fIT_{2}). If there is an one-to-one correspondence between the vertices of T_{l} and T_{2} such that the corresponding edges are equisep arable, then T_{ }and T_{2} are called equiseparable trees. Recently, Gutman, Arsic and Furtula investigated some equiseparable 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 def inition for equiseparability and then prove some results about relation of equiseparability and isomorphism of trees. We also obtain an exact expression for the number of distinct alkanes on n vertices which three of them has degree one.