Are Schlumbergera and Diphylleia both small cacti?
no
Schlumbergera
Schlumbergera is a small genus of cacti with six species found in the coastal mountains of south-eastern Brazil.
 Plants grow on trees or rocks in habitats that are generally shady with high humidity, and can be quite different in appearance from their desert-dwelling cousins.
 Most species of "Schlumbergera" have stems which resemble leaf-like pads joined one to the other and flowers which appear from areoles at the joints and tips of the stems.
 Two species have cylindrical stems more similar to other cacti.
 In Brazil, the genus is referred to as "Flor de Maio " (May flower), reflecting the period in which they flower in the Southern Hemisphere.
Diphylleia
Diphylleia is a group of small herbs in the Berberidaceae family described as a genus in 1803.
 It is native to the eastern United States and eastern Asia.
Schlumbergera orssichiana
Schlumbergera orssichiana is a species of plant in the Cactaceae family.
 It is endemic to a small area of the coastal mountains of south-eastern Brazil where its natural habitat is moist forest.
 It grows on trees as an epiphyte.
 It is in the same genus as the popular houseplant known as Christmas Cactus or Thanksgiving Cactus. Fishhook cactus
Fishhook cactus is a common name for any hook-spined species of the genera "Mammillaria", "Echinomastus" or "Sclerocactus".
 They are small cacti, usually growing up to 6-7 inches (20 cm) high, and are shaped similar to a barrel cactus.
 They are not to be confused with the fishhook barrel cactus ("Ferocactus wislizenii") of the Sonoran and Chihuahuan Deserts.
 The Fishhook cactus is a large category of around 150 species. List of holiday cacti diseases
This article is a list of diseases of Holiday Cacti ("Schlumbergera truncata"). Schlumbergera kautskyi
Schlumbergera kautskyi is a species of plant in the Cactaceae family.
 It is endemic to a small area of the coastal mountains of south-eastern Brazil where its natural habitat is rocky areas.
 It is threatened by habitat loss.
 It is in the same genus as the popular houseplant known as Christmas Cactus or Thanksgiving Cactus. Areole
In botany, areoles are small light- to dark-colored bumps on cacti out of which grow clusters of spines. Areoles are important diagnostic features of cacti, and identify them as a family distinct from other succulent plants.The spines are not easily detachable, but on certain cacti, members of the subfamily Opuntioideae, smaller, detachable bristles, glochids, also grow out of the areoles and afford additional protection. Schlumbergera russelliana
Schlumbergera russelliana is a species of plant in the Cactaceae family.
 It is endemic to a small area of the coastal mountains of south-eastern Brazil where its natural habitat is moist forest.
 It grows on trees as an epiphyte.
 It is one of the parents of many of the popular houseplants known as Christmas Cactus or Thanksgiving Cactus. Schlumbergera truncata
Schlumbergera truncata, the false Christmas cactus, is a species of plant in the Cactaceae family.
 It is endemic to a small area of the coastal mountains of south-eastern Brazil where its natural habitats are subtropical or tropical moist forests.
 It is the parent or one of the parents of the houseplants called Christmas cactus or Thanksgiving cactus, among other names. Echinopsis
Echinopsis is a large genus of cacti native to South America, sometimes known as hedgehog cactus, sea-urchin cactus or Easter lily cactus.
 One small species, "E. chamaecereus", is known as the peanut cactus.
 The 128 species range from large and treelike types to small globose cacti.
 The name derives from "echinos" hedgehog or sea urchin, and "opsis" appearance, a reference to these plants' dense coverings of spines.