What's the explanatory value of metrical trees used to account for prominence relations or syllable stress? At first reflection, it seems to me like rules should be sufficient (indeed, rules and trees coexist in most accounts) - or are there some particular phonological phenomena that the trees are very good at modeling?
The primary problem that metrical trees are intended to solve is the representational problem of what the rule system actually produces. Prior to L&P 77 and as exemplified by the SPE analysis of stress, stress was a feature with scalar values theoretically ranging from 0 to infinity, though in practice values were capped at 5 because of this embarassment of riches. The SPE system also had a general convention that when a rule adds 1 stress to some vowel, all other stress numbers in that cyclic domain are reduced by 1. This system with stress numbers stuck out as a formal oddity, allowing way more possible types of representations than could be empirically motivated in language. There is no segmental analog to the stress-reduction convention, for instance you do not find that making one segment [+coronal] causes all other coronals to become one degree less coronal.
Metrical trees limit the kinds of distinctions that can be made, which from a theoretical perspective is a good thing. The main insight of metrical systems is that language only needs a two-way distinction between "strong" and "weak" (or you may fill in the blank with "prominent", or "stress"). That distinction was combined with a theory of phonological constituency and a theory of locality (which effectively restricted how far you could look up to compute distinctions -- thus "4 stress" was not accessible information). Then you have a classic overgeneration argument: "SPE theory allows you to have all of those classes of rules and representations yet there is only evidence for class A; Metrical theory only allow you to have class A; therefore metrical theory is the superior theory".
The line-drawing aspect of metrical trees is of course just a graphical means of representing certain formal properties, and the subsequence real question was, exactly what claimed properties are justified. The most robust distinction is the prominent / non-prominent distinction, which can be applied at all sorts of levels (i.e. within the syllable; between pairs of syllables; between pairs of pair of syllables...). The second is the concept of "constituency", where in fact there has been significant retreat in the amount of structure postulated (typically just syllable, foot and word). Earlier metrical theory tended to rigorously follow the principle of binary branching to the point that you had to break a string of 8 syllables up into a 4-deep tree grouping of syllables, whereas the facts on the ground never necessitated saying anything more than "syllables group into feet this way" and "this foot is the most prominent in the word" (I will remain officially neutral about the questionable construct "colon" between foot and word).
SPE stress theory heavily exploited the assumption that "phrasal stress" and "word stress" are the same thing. With word stress vastly simplified under metrical theory, the next step, initiated by Liberman and subsequent work by Pierrehumbert, was to account for supposed stress differences at the sentence level, which was done by analyzing the facts in terms of tones in an intonational system. L&P77 does extend metrical representations to phrases by exploiting some of the power of metrical grouping, but when recursive constituency is eliminated as it was, the question arose as to how to deal with what such structures did -- and there is vast literature answering that question.
So, prominence and some theory of constituency are implicit in metrical theory, and are well motivated, but recursive constituency which is also part of classical metrical theory is not well motivated.
I am not a specialist in phonology, so take the following with a grain of salt. But my recollection of metrical trees is that just assuming rules on the liner order of syllables is not enough, because the hierarchical structure of languages does matter for stress assignment as well. And, as L&P show, the metrical structure seems to parallel the syntactic (or morphological structure) quite well.
Also, as L&P point out (if I remember correctly, maybe that was later), their system predicts the systematic ambiguity between broad and narrow focus. So, if you have two expressions a and a, you have two stress pattern: W-S and S-W (i.e. aB and Ab). However, there are three focus configurations: focus on a, focus on a and focus on ab. So one pattern has to be used for both a narrow and a broad focus (for instance, aB).