François Lauze University of Copenhagen In this work we discuss a model of geometric trees and tree-shape spaces developed by A. Feragen and F. Lauze and others I present a construction of the shape spaces, some of their topological properties and discuss metric structures, especially tree-edit distances (TED) and the quotient Euclidean distance (QED), which make it into a geodesic metric space. Copmputation of geodesics for QED, middle points and mean trees are discussed. Applications mainly from Medical Imaging are presented.