On the enumeration of certain sets of planted plane trees

Using the definition of planted plane trees given by D. A. Klarner (“A correspondence between sets of trees,” Indag. Math. 31 (1969), 292–296) the number of nonisomorphic classes of certain sets of these trees is enumerated by obtaining a one-to-one correspondence between these classes and certain sets of nondecreasing vectors with integral components. A one-to-one correspondence between sets of (r + 1)-ary sequences and a certain set of planted plane trees is also established, which permits enumeration of this set. Finally, a natural generalization of Klarner's one-to-one correspondence between the above sets of trees and certain sets of edge-chromatic trees is obtained.