Since haiku have a very limited syllable count, one wants to minimize redundancy. They also tend to emphasize subtle, indirect expression beyond the constraints of length. A haiku's brevity also favors contrast or difference to guide the reader into its sense; when different images are declared true, the reader can form a synthesis that is more true to the meaning than any single extensively portrayed image or collection of similar images. This contrast also has an aesthetic flavor, which may fit with an oriental yin-yang perspective.
Articles (a, an, the) can often be avoided, providing one syllable for more meaningful expression. Prepositions can sometimes be folded into verbs or elided, and conjunctions can sometimes be elided.
For "Mountain" contrast the original:
Mountains under snow
Cold air breeze through the forest
Broad snow-cloaked mountain
Cold sighs, dreaming trees listen
Nature's loud whisper
By replacing "under" with "cloaked" (alternatively "draped" if one wanted a more elegant sense) one provides more flavor ("under" only gives position) and incidentally gains a syllable, which in this example is used to express something of the strength of the mountain ("broad"). The change to singular "mountain" fits the title and increases the emphasis on prominence. Replacing "air breeze" (which has redundancy) with "sighs" both gives a more imaginative, personified sense and reduces length; replacing "through the forest" with "trees" reduces length. The syllables gained allowed the trees to act and to be further described. "Dreaming" was chosen to add a sense of peacefulness (a guess at the intent); "listen" is both passive and active, fitting sleep and awareness. Removing "ing" provides a syllable for "loud", which both reminds of the robustness of the mountain and provides contrast with "whisper" to give a sense that "loud" is not meant as high decibel but to communicate powerful effectiveness in gentle softness (which might remind of "snow").
Forest full of trees
Buzzing like bees, yet silence
A single shadow
Crowded poplars meet
Silent buzzing, no bee sings
One shadow drooping
Replacing "trees" with "poplars" — which may be problematic if the such trees are out of context — provides more definition (poplars also give a hint of a height/pride which excludes) and, incidentally, a word-play with "popular" at the cost of one syllable. The single tree type may also properly or improperly express a sense of sameness or commonality. Replacing "full of" with "crowded" hints at a social context, reinforced with "meet". (An alternative "Proud poplars gather" gives more sound play, reinforces pride — which may not be desired —, and increases the sense of coming together by choice, but removes the abundance of "full"/"crowded".) Replacing "yet silence" with "silent" juxtaposed with "buzzing" frees syllables and strengthens the contrast. Replacing "buzzing like bees" with "buzzing, no bee sings" continues to indicate that the buzz is not from bees (buzz is a very bee-based sound, so "like bees" is somewhat redundant; my assumption was that the liveliness of bees was intended) while also moving to a singular noun (emphasizing isolation), providing a negation, and presenting some sense of melancholy (no liveliness, no singing). Replacing "a single" with "one" frees two syllables; "drooping" was chosen for its soft melancholy feel but "breathing" might be used to hint of living (able to experience emotion, choosing to continue living, or a partial participation with the trees) or "cold, still" (emphasizing painful isolation and a fragile inactivity which could fall to death or wake to life; "cool" instead of "cold" presents more sense of acceptance).
These alternatives are presented to give a sense of how expression can be tightened. Substantial sound-play was sacrificed and the meaning and sense changed, so they should not be taken as recommendations.
With respect to formal requirements, translation into another language and culture, whether of a text or of styles and forms, means choosing what aspects one considers more important to express. Just as a word-for-word translation of a text might reduce literary expression and cultural resonance, so a strict translation of a form and style might result in less appreciation of the original by a casual reader. For haiku, the concept of syllable is not an exact match for the Japanese constraint, but an independent English-language tradition has established it as an expected translation.
Part of the benefit of constraint is in forcing cleverness. This is particularly appropriate for haiku because subtlety, which brevity encourages, is part of the flavor. Adding length is relatively easy, even when avoiding mere padding.
Knowing when and how to bend or even break the rules is a more advance consideration. Reading and writing experience helps, but the choices are often highly contextual, reflecting the author's personality, the subject, and the expected or intended audience.