In the United States there are currently two major varieties of diagrams in use to represent sentence structure: traditional diagrams, used more or less exclusively in junior high school and high school classrooms, and tree diagrams, the most common method used by professional linguists.
The traditional system is no longer as popular as it once was, but it is still to be found in the back of many schoolbooks. Such diagrams are known as Reed-Kellogg diagrams, named for Alonzo Reed and Brainerd Kellogg, who described their method in two books first published in the 1870s: Graded Lessons in English (1875) and Higher Lessons in English (1877). These are the diagrams with slanted lines and other oddly shaped structures extending quaquaversally to the edges of the page that your ninth-grade English teacher may have forced you to draw them if she was a traditionalist. A simple example is shown to the right.
Reed and Kellogg were not the first to represent sentence structure visually, nor did their system immediately replace all others. The Reed-Kellogg method only won out after a period of competition among different schemes. This early history, however, is essentially unknown today. Reed-Kellogg diagrams are a tool of textbooks, and as Thomas Kuhn noted, textbooks tend to efface the history of their subject. Nor have there been many contemporary writers who have taken an interest in the topic. Kitty Burns Florey published a popular account, Sister Bernadette's Barking Dog, which contains modest historical detail, extensive speculation, and some significant errors. A PhD dissertation on the topic also exists (Brittain, 1973), which is an invaluable survey of many early graphical schemes. This work, though suffers from a tendency simply to catalog the various systems. The analysis is confined to considering how adequate the systems were with respect to transformational-generative syntax as it was understood in the early 1970s. Brittain provides minimal historical narrative and does not try to explain the reason for the sudden appearance of diagrams in the middle of the nineteenth century.
The following pages represent notes for an article I am writing on the topic, including a scan of a substantial portion of the earliest attempt to provide a systematic scheme for diagramming, Clark's Practical Grammar (1847). If you have copies of any other 19th-century works that use different diagramming schemes and are willing to send me scans, I will add them here.
I do want to add that I present these systems for their historical interest. I do not recommend that anyone try to use these to understand English grammar directly. They are completely outdated and contain assertions about English grammar that are demonstrably false. Trying to teach yourself grammar from one of these books would be like trying to learn physics from a nineteenth-century textbook. Phlogiston is interesting to study if you want to know how people of the past thought, but it would be silly to cling to it as a way of explaining things now.
Note that the scan of Clark (and any other 19th-century works here) is public domain, but my commentary remains covered by the Creative Commons license.
1. I would like to thank Frank Schumacher for making a copy of Brittain's dissertation available to me.