I was unable to find any real information regarding tournaments for competitive sentence diagramming. However, if you search for "sentence diagramming tournament" and similar phrases, you will find many websites that offer fun sentence-diagramming games and quizzes. Some of these sties seem to have forums, which might mention actual tournaments if they happen in the future.
Yes, there are several websites that allow you to diagram your own sentences. Some popular options include "Web Diagramming Tool" by ReadWriteThink, "Easy Diagramming" by Grammar Bytes, and "Sentence Diagrammer" by Diagramming Sentences.
Tree diagramming
It is called diagramming a sentence.
Schematic or drawing showing the layout of components.
mallory,simon,and jared went on an adventer
A network diagramming technique in which activities are represented by arrows and connected at points called nodes to illustrate the sequense of activities; also called arrow diagramming method (ADM)
A slanted line like this slash: \ See an example here: http://www.english-grammar-revolution.com/diagramming-the-parts-of-speech.html#predicate_nouns
web diagramming
Action verb line.
a web digram is a digram
The purpose of diagramming depends on the application it is used for. Companies diagram their websites to make it easier for users to browse or find different sections. Engineers diagram to plan out there designs and have a better idea, that is easier to follow, laid out on paper.