This project centers around Triangle Y - moves (TY-moves). TY-moves are an operation on graphs (collections of vertices and edges), where we remove the edges AB, AC, and BC of a triangle with vertices A, B, and C, insert a new vertex D, then insert the edges AD, BD, and CD. We can move between several different graphs by doing TY-moves and Y Triangle - moves (YT-moves), and we can organize these graphs by placing graphs with different numbers of vertices above or below each other (as a TY-move adds a vertex). We call this a family diagram. We call a graph G TY-realizable if we can find a family of graphs that has G for a family diagram. We have been going through graphs looking for a way to characterize all trees, whether it is the case that they all are the family diagram of some graph, or that only some types of graphs are TY-realizable. We have categorized all trees with four and fewer vertices as TY realizable as well as some with five or more.