I have just started studying dependency grammar and I am really struggling with the relationship types and trees. I have only ever drawn classic syntactic trees so I keep getting confused. Could you recommend any resources that have dependency grammar tree examples I could use to study. Thank you.
Dependency grammar (DG) is an approach to the study of the syntax and grammar of natural languages that is quite distinct from phrase structure grammar (PSG), which is also known as constituency grammar. The modern history of DG begins with Lucien Tesnière's major oeuvre (1959), whereas the modern history of PSG begins arguably with Noam Chomsky's first prominent work (1957).
DG views linguistic structures in terms of a one-to-one mapping of atomic linguistic units to the nodes in structure, whereas PSG assumes a one-to-one-or-more mapping. The distinction is clearly visible when one compares the tree structures. The next trees are taken from the Wikipedia article on DG:
The DG tree on the left contains seven words and seven nodes, whereas the PSG tree on the right contains seven words but thirteen nodes. Both trees employ the convention whereby the words themselves are used as the node labels in the tree. Note that both trees allow one to identify heads and dependents. Due to the one-to-one mapping of DG, however, the dependencies between syntactic units (words and phrases) are direct, whereas the one-to-one-or-more mapping of PSG often results in dependencies that are indirect and hence less visible in the tree representations.
There are various conventions used to show DG dependencies. The next DG trees are also taken from the Wikipedia article on DG:
One encounters the convention of arced arrows shown in (e) here frequently. The arced arrows are preferred by computational linguists in the field of natural language processing (NLP). DG is preferred over PSG among computational linguists because of the ease with which one can produce DG parses of actual sentences and texts.
Indeed, proponents of DG often point to the simplicity of DG analyses compared to PSG analyses. When I teach DG to students, they can produce dependency parses of many sentences after just an hour of instruction. The same cannot be said of students learning to parse sentences in terms of PSG. The proponents of PSG counter, though, that while DG parses may be easy to produce, they are too simple and hence cannot convincingly shed light on the multiplicity and variation of syntactic phenomena that occur in natural languages.
Concerning the dependency relations (i.e. the "relationship types" mentioned in the question), the inventory of relations varies based on the annotation scheme assumed or the type of DG one adopts. For me, the relevant point about the relations is that one need not focus on them. Dependency parses can ignore them and yet still provide crucial insight into the syntactic structures at hand. Dependency parses show the groupings of syntactic units very efficiently.
Finally, there are indeed various types of DG, as mentioned by Sir Cornflakes. The Universal Dependencies (UD) approach that he links to is one prominent type of DG, and one that is debated. I recommend also checking out the Surface-Syntactic Universal Dependencies (SUD) approach, too, since I think its annotation scheme is more plausible.