Should the edges of landuse=* or boundary=* areas share points with streets that form their borders?

In short: no. While landuse areas can (and probably should) share nodes with other areas they are touching like the boundary of next landuse area or the riverbank way they should not share nodes with a street. The reason is - as you correctely stated - that the trees don't extend into the middle of the street. Once we start to map streets the same way as rivers that is with a centerline way for routing and and closed boundary way to record extend then landuse areas should probably share nodes with that outer way.

Note however that you should not needlessly change the mapping if a user mapped an area using shared nodes between landuse areas and roads. It's a lot of work which doesn't help much but can introduce a lot of subtile problems.

There are different opinions on this, and both methods have advantages and disadvantages. Both methods are "correct", none is "wrong".

A good rule of thumb is:

• If you know the precice boundaries of the object you are mapping then try to include this in the data.
• If you are uncertain of the exact boundary of the object do not make up some arbitrary data.
• Respect the work of others.

That means that if you for instance are mapping a forest from good arial photos, let the boundary go parallel to the road where you see the forest ending and the road area starting. If you for instance have surveyed the area and know that the forest runs up to the road then let the forest and the road share ways.

If someone else has mapped something using one method, do not switch to the other method unless you are actually editing the are and feel you make the switch in order to carry on with your work.

You should not let several ways share too many nodes as this may become hard to edit; if a long road forms the edge of a forest and you want to re-use the road geometry, create a multipolygon relation for the forest and make the road one of the "outer" members of that.

In general, the meadow starts beside the road and not in the middle of it.
In general, the limits of a country do not grow or shrink with the forest.
Etc.
Furthermore people tracing roads and boundaries usually work with a higher degree of precision than landuse which they often meet as long straight lines. No blame but it is unpleasant to have to detach the landuse from everywhere when improving roads.
So, please attach landuse only to landuse and the person who will rework the road will feel like improving your landuse instead of pushing it aside.

This question still seems to be referred to despite its age. The main principle in this matter is to respect the mapping of other users, and that there are numerous different opinions about the correct way of mapping these adjacent features.
In March 2021 the subject of how to map land cover & land use features, in relation to adjacent highways was discussed on talk-gb[https://lists.openstreetmap.org/pipermail/talk-gb/2021-March/026507.html]. There was a wide ranging discussion, but no clear consensus about how the features should be mapped. Where those features border with a highway, the mapper should avoid using shared nodes if possible.

Unlike boundaries, landuse should shared nodes with roads.

Boundaries are administrative data, they cannot be surveyed. Both landuse and roads are physical data, which can be acquired via aerial imagery or local survey. Boundaries and roads may share roughly the same shape at low resolution. But looking closer, you may notice the road has little curves, and you want this data in OSM, without modifying the administrative straight boundary. They are not the same nature, they come from different data source, that is why they should not share their nodes.

Now landuse is a physical property that is heavily influenced by roads (in modern countries), because farm tractors and cows will avoid to cross roads, water cannot flow across a road... Landuse and roads should share nodes if landuse is different on each side of the road, because the road IS the limit of the landuse, and you should not duplicate data. This way, if a road shape is refined or adjusted, the landuse will automatically take advantage of it.

It is true that the trees do not extend into the middle of the road. But roads go on top of landuse. If a road runs in the middle of a single landuse, you will not split the landuse in two different areas. And I doubt we will ever map the roads as areas, because unlike riverbanks, roads have pretty constant width.

Mappers working exclusively on streets and roads may be scared by landuse areas stuck to their roads, because area editing is a significantly more complex task than editing a wired network. Just trust the landuse author: if he/she share nodes with you, he/she wants you to edit the area as you edit your road. However, please note that moving nodes along a straight road segment to the next curve, to improve the curve smoothness without "spending" more nodes, is a dangerous shortcut; you would break the attached landuse. It is better to simply add nodes in curves that need smoothing, and delete useless nodes from straight lines.

Selecting objects sharing the same shape is a bit tricky. If nodes are not shared, you can zoom in as far as necessary until ways separate one from another. This will not work if nodes are shared, but this trick is a bad solution anyway. In JOSM, use the middle mouse button, or ALTGR + left click (Linux), or ALT + left click (Windows), to toggle the object selection between possible ways. In Potlach 2, use the "/" key.

Thanks for reading my point of view. Let me stress that anyway you should respect the work of others. My words do not give you permission to merge roads and landuse anywhere you want. If some mappers want to let the road network float freely over the landuse, let them do it this way, and find another area where nobody is working on the landuse (easy to find!).

Happy mapping!