1

I wrote:

Figure 4 shows the adjoining of a subtree with root S to the projection chain at node V.

Should it be:

Figure 4 shows the adjoining of a subtree with the root S to the projection chain at the node V.

2

Even though they are not capitalized, you can think of "root S" and "node V" as proper nouns (naming specific things), and as such no article is required. But either option is possible.

Figure 4 shows the adjoining of a subtree with root S to the projection chain at node V. (proper nouns)

or

Figure 4 shows the adjoining of a subtree with the root named S to the projection chain at the node named V. (common nouns)

named is added for clarity but can be omitted.

1

Your original sentence is better:

Figure 4 shows the adjoining of a subtree with root S to the projection chain at node V.

If your audience is already aware of the roots and nodes, referencing them with "the" sounds a bit odd.

1
  • 1
    I agree with this answer. This style is perhaps a bit specific to mathematical or scientific writing, but in that context the first is definitely the most natural.
    – Tom Church
    Feb 26 '18 at 6:31

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