Long distance dependencies have to do with clauses. They occur when some element within a clause is "missing". Depending on what kind of grammar you subscribe to, you might want to think of that missing element as a 'gap'. Consider the following:
The 'sentence' above will seem odd if not ungrammatical to most English speakers because the verb like usually requires an Object. Such an Object is missing from the phrase above. We cannot use like with its normal meaning without a Direct Object in a straightforward clause:
A: Do you like chocolate?
B: *Yes, I like. (ungrammatical)
However, there are more complex constructions where we do see strings like I like without any overtly expressed Direct Object:
That is the elephant I like.
In the sentence above we can see that there is a 'gap' in the clause I like where we would normally expect a Direct Object to be. We can model the sentence like this:
That is the elephant [I like ___ ].
A pertinent question at this point might be: How do we interpret this gap so that the clause makes sense? Well, our interpretation of the gap here is tied in with the 'antecedent' of this relative clause, the elephant. We can liken the relationship between this gap and the antecedent to the relationship between a pronoun and its antecedent:
That is the elephant(i). I like it(i).
Here the subscript (i)'s are there to show that our interpretation of the word it is co-indexed with the word elephant. They refer to the same thing. So now we can model our relative clause sentence like this:
That is the elephant (i) [ I like __(i) ].
Now we can see from the above that our interpretation of the gap in the clause "I like _" is anaphorically dependent upon our understanding of the elephant. There are many such constructions in English (and of course other languages) where our interpretation of the identity of some missing element is anaphorically dependent upon some other word or phrase. In many cases this is because the missing element has 'moved' to the left of the body of the clause:
Now I like that elephant.
Now that elephant(i), [ I like __ (i) ].
You like which elephant?
Which elephant (i) [ do you like __ (i) ]?
But in other instances one might feel that the feature is just a part of the construction:
Bob (i) is easy [ to find __ (i) ]
(In the sentence above we might not feel that Bob has moved to the beginning of the clause from another position, exactly. It is more as if a second Bob has been deleted from the end of the smaller clause. Again, this would depend on what kind of grammar you subscribe to.)
Now it is the relationship between these gaps and the words or phrases that they depend on for their interpretation which we refer to as long distance dependencies. Why do we call them long distance dependencies? Well, for one thing, these gaps are non-adjacent to their antecedents. We could suppose that we could measure the distance between the antecedent phrase and the gap itself to see how long distance this dependency was. So consider the following sentences:
- Who (i) did Bob say I punched __ (i) ?
- Who (i) did that awful nit-picking man with the bobbly hat and stripy shoes punch __ (i)?
In (5) there is a five word interval between the gap and the antecedent phrase. In (6) the interval is fourteen words. However, this kind of interval is of little interest to most linguists, and has little to do with the term long distance dependency. In fact it is sentence (5) and not sentence (6) which we would say best illustrated a long distance dependency in its most important sense.
In all the other examples given so far, the antecedent phrase occurs directly to the left of the main body of the immediate containing the gap. So if we showed this in (6) we would get:
Who(i) [did that awful nit-picking man with the bobbly hat and stripy shoes punch __ (i)]?
Here we see the word who occurring directly to the left of the nucleus (the main body) of the clause containing the gap. The long string that awful nit-picking man with the bobbly hat and the stripy shoes is just the subject of the verb punch. Because it is long it has moved the gap further away from its antecedent who. But nonetheless, the word who occurs here directly behind the very clause containing the gap. We can compare that to (5):
Who (i) did Bob say [ I punched __ (i) ]?
Now here we see that the clause I punched __ is embedded within a larger clause Bob said [ I punched __ ]. So here the gap depends on a word not just to the left of the immediate clause it occurs in, but on a word to the left of a different larger clause that that smaller clause is embedded in. So the dependency relationship here is spanning two clauses. It is this property of the antecedent gap relationship existing across clause boundaries which leads us to call them long distance dependencies. Some linguists call constructions such as the ones above unbounded dependency constructions, because in theory there is no limit to the number of embedded clauses involved:
- That's the elephant Chris said Ben believes you like.
- That's the elephant (i) [Chris said [Ben believes [ you like __ (i) ]]].
In reality, with many of these constructions there are practicalities which make multiple embeddings difficult to process, for which reason sentences like (7) are relatively rare.
So, if we wanted to somehow measure how long-distance the dependency between the gap and its antecedent was in a particular sentence, we would most likely want to quantify this in terms of the number of embeddings involved—in other words the number of clause boundaries intervening between the gap and it's antecedent +. The number of words is usually of less interest to us. If we did want to measure the intervening material though, we might want to measure it in terms of syllables, not letters, characters, words, or phrases.
+ When measuring this it would be important to note that we'd only want to count clauses that the gap actually appeared in. A whole discrete clause occurring between the antecedent and the gap would not count for anything.
With regard to the Original Poster's examples:
1. a two-hundred-and-fifty-old, tall, fragile tree
We wouldn't regard this noun phrase as exhibiting any kind of LDD. One reason for this is that there is no clause involved. The second reason, however, is that there is no gap in the phrase. Every element of the noun phrase is fully expressed and in exactly the position we would normally expect it to be.
2. The man that I saw yesterday after lunch went fishing.
This sentence on the other hand does use an LDD construction. However, there is no gap here between the word lunch and the verb went. The word lunch is just the last word in the noun phrase the man that I saw yesterday after lunch. This noun phrase is occurring in the normal subject position in relation to the verb went. So there is no missing element between the two. However, within this subject noun phrase there is a relative clause that I saw yesterday after lunch. This clause has a gap where we would expect the direct object to be. The gap is co-indexed with the antecedent the man:
3. The man(i) that [I saw __(i) yesterday after lunch] went fishing.
Again, how you wished to measure the 'distance' of the dependency relationship would depend on what exactly you were trying to show.