(Hope I'm not too ninjaed, the over-long message that follows is full of a lot of principles I could suggest, and I had to vacate my workstation for a few hours in the middle of it. It is of course a whole lot of thoughts about possibilities, not "this is how it should be done" in any firm way.)
Height-wise, I think MTT would probably have the aforementioned 'crown', essentially a sapling[1], continuing while-ever the tree is 'happy' and has space to expand into (being shaded by an overbuilt structure/other trees should make it less happy, in the first place (although a tile with light let through might encourage it to grow tall, through that gap), and restrictive the growth anyway. With appropriately tuned elven tree-house building practices (actually nurtured growth of the tree itself), this could be how Elves create/maintain their tree-houses.
Width-wise, I see 'trunk' tiles (see footnote) slowly (according to species growth-rate, but over years) butting in on adjacent squares (still centred on original square[2]), according to free space within two tiles (for normal growth) or even one tile
(stunts growth, but will fill gaps, eventually, assuming further up is not restricted). If there could be a similar 'partial-fill' tile, as per the vertically-expanding 'sapling' top, it would be nice (still traversable, not "trunking up" until it that tile actually matures another stage or three further, never instantaneously crushing a creature that passes, but possibly at some point suffocating anything unable to move off of that tile for extended periods, yet not actually dying of any other problem).
That's width-wise at the base. At some point of vertical growth, there'd be branch outgrowths (passable tiles, also respecting the vicinity of other structures/tree branches from other trees), and as the trunk below goes beyond a certain level of maturity and the branches extend further, there'd be further trunk-outgrowth (again, respecting nearby tiles). I could see outgrowth impinging upon constructed (or grown, see below) steps, but not actual walls. (The point where a wall is collapsed by an expanding tree-trunk seems a bit difficult to sort out, and a cliff-edge tree causing landslips, especially if expanding root-systems are also implemented, would be nice but... problematic.)
It may be possible to get 'bowls' in the tree, and I imagine Elven tree-houses might make use of these to form their larger rooms (in sufficiently old trees), as the growing trunk splays out, instead of up, and around, and perhaps over the top again. Probably happens naturally, to some extent, but Elven interferenceguidance and nurturing in the tree-growth would be the obvious way to get this.
Unless elves generate 'stairway' branch formations (should be possible), then [CLIMBING:TREE]-tagged creatures (or anything tagged with a superset [CLIMBING:ALL] tag or something also inclusive) should be the only ones to actually access the upper levels. I could imagine equipment (footspikes, also useful in mêlée-kicking!, but slows normal ground movement while worn, and a carried rope, perhaps) that could be used (in conjunction with the woodcutting skill?) to allow movement by non-intrinsic creatures. Or you could just build your own stairway access.
For felling, I'd say it requires that the tree has at least one impassible 'trunk' tile at ground level (or, with suitable access, at the level you're cutting the tree off from), to designate as a target for a woodcutting job. If it's the sole 'trunk' tile, it'll cause a cave-in, but perhaps the woodcutting job can either be re-arranged from simple tile-designation to require drawbridge/target-like 'direction' specification on initial 'designation', or you get a chance (for at least sufficiently competent woodcutters) to specify the direction of felling with a play-freeze and action-centring dialogue at the moment the job is started. Training might be on trees small enough to have no woody top (no questions get asked, but no danger), but tackling a taller tree would be dangerous for new woodcutters, who would have a degree of chance/self-preservation in their choice of how to undergo the job. Or if not for themselves (it always falls away from the tile they finish tackling the job from?[3]), in making sure that it falls in a direction no 'likely' to cause (unintentional) damage to any others. I foresee the above levels disassembling into individual logs, spread a number of tiles away in the direction of felling, according to how high above they were in the first place, but perhaps an axle-like 'single trunk' could be produced, give or take stress-induced fracture. (It'd be nice to be able to fell a tree across a gap, and use the top of it as a walkable bridge, albeit with a chance of it falling wrong, or even falling while being traversed.)
Multi-width-tiled treeswould need each trunk-type tile to be 'mined'/woodcut. It'd be nice for some form of stress system, so that a suitably heavy tree-top would "snap" a single (or couple) of remaining 'stump' bark, again mean care when felling a large-enough tree, but as cave-in mechanics for rock aren't in at the moment, I wouldn't press for that. Building into 'hard' wood would perhaps be consider sufficient support, under current circumstances, a bit like a rocket-gantry. Basically the current tile-supporting rules would apply, but only for the actual "trunk" wood, not mere branches or leaf-ends.
Yes, branches and leaf ends could probably also be pruned/trimmed (or damaged/built over by structures)... It could be done aggressively. Shorning a tree of its 'greenery' would leave a lifeless tree-trunk (or at least mean that it would take a lot longer for new growth, and it would need that new non-trunk growth before it could expand its trunk, upwards or sideways, again). Or it could be done more delicately, allowing harvesting of wood. I think the practice of Pollarding should be possible, or at least given an in-game analogue. This would make available "stick" wood, as opposed to "log" wood.
So, different species of tree would have different ratios of height-to-width growth, as well as different tendencies for maximum heights and widths, giving different profiles (unencumbered by neighbours or geological/constructed features 'in their way', light level obscurations, etc), and different responses to the aforementioned constrictions. I think it should be much rarer that a tree (or tree-analogue) block a 1-wide corridor than it is now, especially with a low roof but perhaps (given time) it could still do if left otherwise undisturbed. I also can't imagine a 'wall' of trees, although a stand of trees would have narrow gaps. If 'branches' fill them then it slows, but does not prevent, movement of creatures and even liquid.
As to representation. I foresee a trunk being like a construction "brown coloured", or whatever best matches in the 4-bit colour-scheme (going on current default graphics), but different trees may have different colours. A 1x1 section of trunk would be an "O" of that colour, but perhaps a greater-width trunk (more than one tile) would use the ASCII-tabular set (single line, rather than tramline) as the outline. Brown/Green-tinted "shaded tile" characters of various degrees of saturation could represent the (at least partially) 'passable' tiles with branches and leafy twigs. Something sapling-like might be seen at the top. Or at the top of every currently active crown for a given tree.
Is that enough speculation? Probably repeating someone, now, or contrasting violently with them, but just a few ideas I've had (possibly some that I've picked up from elsewhere, so I can't credit/blame myself entirely for all of them).
[1] Or several stages. Starts off sapling on ground, becomes young tree/tree-top on ground, becomes trunk on ground and sapling on Z+1, perhaps, then on Z+1 becomes yt/tt, before becoming more trunk with sapling/whatever on Z+2.
[2]..9..
.716.
.304.
.528.
.....
...but that'd be 9x several years each of continual growth
[3] To mirror RL, they start on one side, perhaps, then finish off the cut on the other side. Needs access to both sides, of course. Diagonally-adjacent tackling of the final cut would be possible, but dangerous. Cutting a tree now set in a corridor (that blocks the opposing side) would be pretty much lethal (at least for non-high skills in the task), and perhaps that's why you'd construct a means to access the higher levels and cut them down first, only removing the ground-level 'stump' trunk when there's no cave-in danger)