This thesis is a study of harmonic maps of trivalent trees into Euclidean space. The existence of such maps is established, and uniqueness is shown to hold up to a certain isotopy condition. Moreover, within its particular isotopy class, each harmonic map is shown to be a local minimum for the energy functional. A harmonic map of a trivalent tree is determined by its associated nodes. Collectively, these nodes are a function of the lengths of the parameter spaces of the paths which comprise the map. It is shown that this node function can be continuously extended to certain parts of the boundary of its domain; these parts of the boundary are closely related to the geometry of the trivalent tree which serves as the domain of the given harmonic map.