
phyllotaxis (also, phyllotaxy) [ fil-uh–TAK-sis ] noun: the manner in which leaves are arranged with regard to the axis and in relation to one another; the study of the factors that determine the growth patterns and arrangements of plant leaves

The three major patterns of phyllotaxis are alternate, opposite, and whorled, as shown in the illustration to the right.
How leaves are arranged on the stem (axis) is one of the tools for classifying and identifying plants. The word phyllotaxis comes from two Greek root words, phyle, meaning tribe or family, and taxis, meaning arrangement or order. Taxis is also at the root of the word taxonomy, which is the study of the general principles of scientific classification of plants and animals into ranks like order, family, genus, species, subspecies, or variety.
Many a master gardener working the Help Desk has encountered the need for this tool when trying to determine whether an ailing small-leaved evergreen shrub is Buxus sempervirens (boxwood) or perhaps Ilex crenata (Japanese holly) or southeast native Ilex vomitoria ‘Nana’ (dwarf yaupon holly) or some other holly variety or cultivar. Boxwood leaves are opposite, the holly, alternate.

Phyllotaxis is also used as a tool in identifying trees, separating them out into their genera. The most common leaf arrangement found on trees in Virginia is alternate. Three common exceptions having opposite leaves are maple, ash, and dogwood, giving us the MAD acronym as a helpful mnemonic. So you would expect dogwoods to have opposite leaves, like Virginia’s state flower and tree, Cornus florida (flowering dogwood). But even exceptions have exceptions, and unlike other native dogwoods, Cornus alternifolia (alternate-leaf dogwood), as you would guess by its epithet and common name, has alternate leaves. (Note that the Virginia Tech Dendrology site says that the leaves of Cornus alternifolia may appear opposite or whorled where they cluster at the branch tips.)

Photo © Katja Schulz CC BY 2.0
Trees with whorled leaf arrangement are rarer. The leaves of Catalpa speciosa (northern catalpa), native to eastern North America including the Carolinas, grow in whorls of three leaves per node, which helps differentiate them from the invasive exotic Paulownia trees that have similar looking large leaves growing in opposite pairs. (Note that the Virginia Tech Dendrology site says that Catalpa speciosa can have either whorled or opposite leaf arrangement, so for this tree, leaf arrangement is apparently not a foolproof identifier.)
Among herbaceous plants, one of the clearest examples of whorled leaf arrangement is Veronicastrum virginicum (culver’s-root). Although Monarda punctata (spotted beebalm) has opposite leaves, its showy flowers and subtending bracts (modified leaves) are sometimes described as whorled but they are arranged in verticillasters or false whorls. Oclemena acuminata (whorled wood aster or mountain aster) has alternate leaves, one per node, but at the apex of the plant, the nodes are so close together, they give the appearance of being whorled, hence one of the plant’s common names. Another plant that might look whorled is the shrub rhododendron. It, too, actually has alternate leaf arrangement.

Photo © Mary Free

Some sites, like that of the North Carolina State Extension Gardener Plant Toolbox, have detailed plant trait information, including leaf shape and arrangement. (NCSU’s is near the bottom of its plant information pages.) Like colors and textures, these structural factors are very useful when trying to identify plants.
References
Tree Identification, Virginia Forest Landowners Program, Virginia Tech.