Bijections for 2-plane trees and ternary trees

According to the Fibonacci number which is studied by Prodinger et al., we introduce the 2-plane tree which is a planted plane tree with each of its vertices colored with one of two colors and -free. The similarity of the enumeration between 2-plane trees and ternary trees leads us to build several bijections. Especially, we found a bijection between the set of 2-plane trees of n + 1 vertices with a black root and the set of ternary trees with n internal vertices. We also give a combinatorial proof for a relation between the set of 2-plane trees of n + 1 vertices and the set of ternary trees with n internal vertices.