Kingdom and Division
Crabapple trees are of the kingdom plantae (plants). They are of the division magnoliophyta, which classifies them as flowering plants.
Class and Order
Magnoliopsida is the class that crabapples fall under, allowing those who see the phrase to know that the seeds of the plant have two leaves. They are of the order rosales, which further narrows down the category under which the trees fall.
Family and Subfamily
Crabapple trees are part of the family rosaceae. They are closely related to roses, raspberries, strawberries, pears and rowan trees. They are part of the subfamily maloideae or spiraeoideae.
Crabapples belong under the tribe maleae. This further classifies them under linnaean taxonomic ranks.
All crabapple trees fall under the genus "malus". This genus is comprised of nearly 40 different deciduous trees and shrubs. Deciduous means that they lose their foliage in the winter.
The species name depends upon which type of crabapple tree is being observed. One well known example is "m. sylvestris sieversii". Sylvestris sieversii is the species name.