The small ground sloth, Hapalops, enters the LRT basal to two extant tree sloths

Hapalops ruetimeyeri
(Ameghino 1887; Miocene; 1 m length) is a smaller ground sloth, basal to tree sloths in the LRT. 26 species has been named. Note the reduced medial and lateral fingers and toes. This informs the digits remaining in the three-toed sloth, Bradypus.

Bradypus tridactylus
(Linnaeus 1758) is the living three-toed sloth, derived from a sister to Hapalops, Bradypus was much smaller and arboreal. The skull is relatively small. The cervicals are long with two more than typical for mammals. Digits 1 and 5 are absent on both the manus and pes. The pedes are turned inward, so the sloth walks, when it has to, on the sides of its feet. Bradypus hangs inverted from trees. It is an herbivore with a slow metabolism.

Choloepus didactylus
(Illiger 1811) is the living two-toed sloth, derived from a sister to Hapalops and Bradypus. The canine teeth are large and sharpened against lower teeth, similar to hippos. Apparently two fewer cervicals are present. The jugal and squamosal angle ventrally. Choloepus also hangs inverted from trees.

According to Wikipedia,
“The two groups of [tree] sloths are from different, distantly related families, and are thought to have evolved their morphology via parallel evolution from terrestrial ancestors.”

Just the opposite, according to the LRT.
The two extant tree sloths are both derived from Hapalops.

References
Ameghino F 1887. Enumeración sistemática de las especies de mamíferos fósiles coleccionados por Carlos Ameghino en los terrenos eocenos de la Patagonia. Boletín del Museo de La Plata 1: 1–26.
Linnaeus C 1758. Systema naturæ per regna tria naturæ, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus I. Editio decima, reformata. McAfee RK and Naples RK 2012. Notice on the occurrence of supernumerary teeth in the two-toed sloths Choloepus didactylus and C. hoffmanni. Mastozoología Neotropical, 19(2):339-344, Mendoza.
Naples VL and McAfeee RK 2014. Chewing through the Miocene: an examination of the feeding musculature in the ground sloth Hapalops from South America (Mammalia: Pilosa). [v1; ref status: indexed, http://f1000r.es/2pu%5D F1000Research 2014, 3:86
(doi: 10.12688/f1000research.3282.1)

wiki/Bradypus
wiki/Hapalops
wiki/Sloth