Are they only used in teaching English in the United States, or in other countries as well?
Sentence Diagramming in the United States
In the U.S. educational system, sentence diagramming was a common technique for explaining the structure of sentences for more than a century. In a Google Books search, one of the earliest instances of sentence diagramming occurs in Stephen W. Clark, A Practical Grammar: In which Words, Phrases, and Sentences Are Classified According to Their Offices, and Their Relation to Each Other (1847). Clark’s system used a series of lozenge-like balloons to accommodate the various pieces of the diagrammed sentence.
Alonzo Reed & Brainard Kellogg, Higher Lessons in English (1878/1885), followed by Jonathan Rigdon, Analysis of the Sentence With Diagrams (1887), adopted a somewhat simpler line-based presentation of the diagramming; and Henry C. Edgar, Sentence Analysis by Diagram (1915) made the presentation cleaner still. Edgar’s format is essentially the one elementary-school teachers were using in Texas (and much of the rest of the United States) during the 1960s when I was a student.
Sentence diagramming continues to be promoted as a useful pedagogical technique in recent books such as Cindy L. Votto, Grammar by Diagram, Second Edition (2003). This particular book is from a Canadian publisher, but I don’t know whether it sells to a Canadian audience.
Sentence Diagramming Outside the United States
In “Sentence diagramming,” a January 1, 2014 blog post on Language Log, Dick Hudson writes:
For a long time 'sentence diagramming' flourished throughout the American school system, and although it was strongly condemned as a useless waste of time in the 1970s, it still persists in many schools. Not only that, but it spread well beyond the USA, so a very similar system is still taught in many European countries (though not, alas, in the UK); for example, schools in the Czech Republic teach sentence diagramming so successfully that researchers are investigating the possibility of including school children's analyses in a working tree-bank of analysed sentences.
One language teacher responding to the post describes using sentence diagramming in Canada and Japan:
I learned Max Morenberg style sentence diagramming during my undergrad (1996), and it has been invaluable to my career as a successful language teacher (Latin, French, English and Japanese) in Canada and Japan, at level ranging from primary to university.
On the other hand, commenters from South Africa and the UK noted that they had never seen tree-style diagramming used in sentence analysis.
A commenter named Nava in a 2008 Straight Dope thread on “Diagramming sentences” reports:
The way we diagram sentences in Spain is completely different from the way it's done in the US, as I learned in a previous thread which my search-fu can't find. Our way [a tree structure] doesn't deconstruct the sentence completely, like yours does. When we diagram a sonnet it stays a sonnet.
According to Mike Guest of Miyazaki University, in “10 Dumb Things That English Teachers in Japan Do” (February 18, 2012), one dumb thing is to “use unproductive teaching methods”:
You know what I mean. The old adage that high school teachers have to teach grammar explicitly by having students diagram and memorize sentence patterns at the expense of dealing with content and meaning-- the result being that students have only receptive, analytical skills and can't use English productively and meaningfully.
So, apparently, at least some university-level teachers are using sentence diagramming in Japan, and others are disapproving of the practice.
Though sentence diagramming appears to refer to more than one thing, it seems to have originated in the United States. Some people do teach it (or a related form of parsing) in other countries, but Australia is certainly not the only English-language country where it is not widely used.
Well, I've been applying Sentence Diagramming (renamed as "Sendi" ) in Thailand, setting it up as a curriculum being taught in an international school for almost 5 years and currently running my own tutor school mainly emphasizing only instructing Sendi to students to attack ERROR questions. To master sendi skills needs the chemistry of observing, collecting, analyzing and hypothesizing---The four essential elementary of being Sherlock Holmes.