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An alternate world is populated by mainly intelligent trees and other plants. These plants are almost exactly the same, biologically, as Earth's plants. The world's climate and geology are very similar to how Earth would be if intelligent life (other than the plants) had never evolved. Without causing any major changes, would it be theoretically possible for these plants to communicate? And how would it work?

Sound

They would produce sounds in one of two ways (probably both)

• By shifting their branches and leaves in just such a way that when the wind blows it makes the sounds they desire. Much like our vocal cords. The drawback is we produce our own wind whereas they would need to rely on air passing through their foliage.


• By creaking. They shift their entire bodies to extremes causing them to make sounds. The added advantage of this is that the young who have "poor language skills" or "developing language skills" are mostly struggling with the fact that they are extremely flexible and don't make a lot of noise as such.

All of these sounds can be detected as vibrations they pick up in their bark or their foliage (or both.)

Earthly Vibrations

Or perhaps they have the ability to cause vibrations in the ground by wiggling their roots. The other trees can also sense the vibrations through their roots. This would like make it hard to make out more than one "speaker" at a time.

Chemical Signals

Similar to the way ants communicate. However this would have to be transferred in some way. Perhaps pollen? Then it is more like they write messages in their pollen and wait for the bees to deliver the messages for them.

Plants already communicate, we simply ignore most of the mechanism they use.

An African tree has been found capable of communicating with its neighbors to warn them about excessive consumption from antelopes, so that they can secrete more tannine. (paper cited below)

Tomatoes communicate using roots: plants growing close to an infected one started producing antibodies against the pathogen. (paper cited below)

Some pines have been found exchanging electric signal, even though they lack neurons. (paper cited below)

Here is a set of scientific papers where you can find more info:

• Baldwin IT, Schultz JC. Rapid changes in tree leaf chemistry induced by damage: evidence for communication between plants. Science 1983;221:277-9.


• Dudley SA, File AL. Kin recognition in an annual plant. Biol Lett 2007;3:435–8.


• Mousavi SA, Chauvin A, Pascaud F, Kellenberger S, Farmer EE. Glutamate receptor-like genes mediate leaf-to-leaf wound signalling. Nature 2013;500(7463):422-6.


• Ramakrishna A, Giridhar P, Ravishankar GA. Phytoserotonin, a review. Plant Signal Behav 2011;6:800–9.


• Robbins CT. Role of tannins in defending plants against ruminants: reduction in dry matter digestion? Ecology 1987;68:1606-15.


• Roshchina VV. Evolutionary considerations of neurotransmitters in microbial, plant, and animal cells. In Microbial endocrinology. Lyte M et al. (Eds), p. 17-52, Springer 2010.


• Simard SW, Beiler KJ, Bingham MA, Deslippe JR, Philip LJ, Teste FP. Mycorrhizal networks: mechanisms, ecology and modeling. Fungal Biol Rev 2012;26:39–60.


• Song YY, Zeng RS, Xu JF, Li J, Shen X, Yihdego WG. Interplant communication of tomato plants through underground common mycorrhizal networks. PLoS One 2010; 5: e13324.


• Van Hoven W. Mortalities in Kudu (Tragelaphus strepsiceros) populations related to chemical defence of trees. Rev Zool Afric 1991;105:141-5.


• Van Hoven W. The tree’s secret weapon. South African panorama 1985;30:34-7

You don't have to speculate very far at all; it's a widely accepted notion, since Prof. Suzanne W. Simard's 1997 research paper, "Reciprocal transfer of carbon isotopes between ectomycorrhizal Betula papyrifera and Pseudotsuga menziesii" that plants communicate using chemical secretions, primarily through their root systems.

Her team proved this by injecting the root systems with radioactive "tracer" isotopes, the passage of which could be tracked through to the root systems of neighbouring plants. It's become known colloquially within the Plant Sciences domain as the "Fungal Internet".

https://interestingengineering.com/study-reveals-plants-communicate-through-root-secretions

https://e360.yale.edu/features/exploring_how_and_why_trees_talk_to_each_other

http://centaur.reading.ac.uk/48102/1/Gorzelak%20et%20al%202015%20Inter-plant%20communication%20mediated%20by%20mycorrhizal%20fungi.pdf

Original paper (requires login for full paper):
https://www.cambridge.org/core/journals/new-phytologist/article/reciprocal-transfer-of-carbon-isotopes-between-ectomycorrhizal-betula-papyrifera-and-pseudotsuga-menziesii/05B4F05708AA2EF3EFA87965EADFE307

When elephants eat the acacia tree, they start down wind because the attacked plant produces toxin as well as pheremones to alert nearby trees so they can start producing toxins.

By starting down wind the elephants can work they way up wind without the trees alerting the other trees

See Alarm Scents

Since both the communications of plants to warn of pathogenes, predators and to exchange nutriant has been mentionned in previous answers, let's just add that Darwin had speculated about the roots being to plants what the brain is to animals source

“It is hardly an exaggeration to say that the tip of the radicle thus endowed [with sensitivity] and having the power of directing the movements of the adjoining parts, acts like the brain of one of the lower animals; the brain being seated within the anterior end of the body, receiving impressions from the sense-organs, and directing the several movements.”

If we have intelligence then you might want to have settlements, which with network effect multiplies communication. Huge root networks like Pando, the largest organism on Earth would enable very fast communication.

Also, since you mention populated by mainly intelligent trees and other plants, we could assume some form of animals might be present and could have been domesticated for communication when roots or chemicals cannot do it.
Based on Earth insects would most likely be domesticated and used for communication, but possibly bigger animals since they are already used on Earth for for seed dispersion on long distances(extract below, source, they could be used on your planet for long distances physical artefacts deliveries.

"large herbivores are irreplaceable as seed dispersers because, relative to smaller frugivores, they are able to consume larger seeds and deliver many more seeds per defecation event over longer distances."

If you want to take it slowly:

Leaves are the mouths

Roots are the ears

So a conversation would involve the tree creating a chemical composition in the leaves to communicate some 'message'. These leaves fall and decompose, with the resultant message being transported through the soil and taken into the roots of another tree.

Naturally, with the dependence on leaf scatter patterns, prevailing winds and distances between trees, such exchanges in conversations take an extremely long time. Hence rather than each leaf containing a single message such as "It's a nice autumn we're having", leaves contain partial messages with a high level of redundancy across the leaves so that a cascade of falling leaves communicates an entire concept or state, in a very similar way as the logograms in the book Story of Your Life (film: Arrival).

As the entire knowledge of one tree can be transmitted in this way, it leads to an accumulation of understanding in large dense forests. Some of which have evolved in ways that would not be expected of 'normal' plant life.