Encoding and decoding algorithms for unlabeled trees

Trees considered in this article, in which at least two receivers emanate from each internal vertex, are found in the works by E. Schroder, R. Stanley, O.V. Kuzmin, etc. Encoding and decoding algorithms for unlabeled planar rooted trees with a given number of end vertices, root receivers, and a sequence of degrees of internal tree vertices during depth-first search are built. Encoding is done in non-decreasing tuples, by assigning labels to the internal vertices of the tree. The proposed algorithms made it possible to prove the existence of a one-to-one correspondence between the studied set of trees and the set of non-decreasing tuples. To find the cardinality of a set of tuples, we used an approach based on the generalized Pascal pyramid.