A complete proof of The Graceful Tree Conjecture using the concept of Edge Degree

The Graceful Tree Conjecture claims that every finite simple tree of order n can be vertex labeled with integers {1, 2, ...n} so that the absolute values of the differences of the vertex labels of the end-vertices of edges are all distinct. That is, a graceful labeling of a tree is a vertex labeling f , a bijection f : V (Tn) −→ {1, 2, ...n}, that induces an edge labeling g(uv) = |f(u)− f(v)| that has a special property g : E(G) −→ {1, 2, ..n − 1} is a bijection. The Graceful Tree Conjecture claims every tree has a graceful labeling. We prove here that every tree has a graceful labeling.