Before revision:  As a point set, this space comprises the previously developed Billera-Holmes-Vogtmann (BHV) tree space while its geometry is motivated from the edge-product space. As the latter, our new wald space also involves disconnected forests, it does not contain certain singularities of the latter, though.
After revision:  As a point set, this space comprises the previously developed Billera-Holmes-Vogtmann (BHV) tree space ; it also contains disconnected forests, it does not contain certain singularities of the latter, though.