On planes trees with a prescribed number of valency set realizations

We describe valency sets of plane bicolored trees with a prescribed number of realizations by plane trees. Three special types of plane trees are defined: chains, trees of diameter 4, and special trees of diameter 6. We prove that there is a finite number of valency sets that have $R$ realizations as plane trees and do not belong to these special types. The number of edges of such trees is less than or equal to $12R+2$. The complete lists of valency sets of plane bicolored trees with 1, 2, or 3 realizations are presented.