Fully Cordial Trees

For a graph \(G = (V, E)\) and a coloring \(f : V(G) \to \mathbb{Z}_2\), let \(v_f(i) = |f^{-1}(i)|\). \(f\) is said to be friendly if \(|v_f(1) – v_f(0)| \leq 1\). The coloring \(f : V(G) \to \mathbb{Z}_2\) induces an edge labeling \(f_+ : E(G) \to \mathbb{Z}_2\) defined by \(f_+(xy) = |f(x) – f(y)|\), for all \(xy \in E(G)\). Let \(e_f(i) = |f_+^{-1}(i)|\). The friendly index set of the graph \(G\), denoted by \(FI(G)\), is defined by
\[
FI(G) = \{ |e_f(1) – e_f(0)| : f \text{ is a friendly vertex labeling of } G \}.
\]

In this paper, we determine the friendly index set of certain classes of trees and introduce a few classes of fully cordial trees.