Toby Jackson

It seems like a simple question, right? We have all climbed a tree and know perfectly well that the branches are the bits that stick out, and the trunk is the bit in the middle. However, when it comes to giving a robust definition that holds for all types of trees, I am having some difficulties.

The problem

It seems to me that there are essentially two types of trees in theory: the ‘fractal tree’ and the ‘Christmas tree’, these correspond roughly to the idealized broadleaf and conifer. Most toddlers know which bit is the trunk for either of them. The problem is that real trees lie somewhere between these two types, which makes it more of a challenge. To make this clear I’ll need to take a step back and explain the context, so bear with me.

Context

I work with terrestrial laser scanning (TLS) data, which is a relatively new technology that can produce beautiful 3D cylinder model trees. Now that we have these 3D cylinder model trees, we want to ask questions like: ‘How much of the mass is contained in the branches and how much in the trunk?’ and ‘Is the tree symmetric about the trunk?’

Clearly, to answer these kinds of questions we need to be able to reliably define which bit is the trunk. Taking this one step further, we would also like to be able to define the branching order. This is just an extension of the same problem: if the trunk has branching order 1, then the next branches have order 2 and then 3 and so on. This would allow us to explore questions about the resource use and the mechanics of the tree. As far as I can see there are two possible definitions.

Definition A

As usual, Leonardo da Vinci has had his say on the matter, so we’d better put his views first. He reckons that a tree is like an upside-down river: “All the branches of a tree at every stage of its height when put together are equal in thickness to the trunk… All the branches of water at every stage of its course, if they are of equal rapidity, are equal to the body of the main stream”. That’s a very compelling idea, and leads to a clear definition of branching order: The branching order increases by one at every branching event.

This definition fits very well with our fractal type tree, but it gives silly results on the ‘Christmas’ tree. If you want to see the problem with this definition just have a scroll up and look at the banner image – every time a branch comes off, the main stem increases in branching order. That means that the trunk stops at the height of the first branch! Despite this obvious limitation, this is the definition which is often used because it doesn’t require any arbitrary decisions and is completely reproducible, which is an important point if we are trying to compare the work of multiple studies.

Definition B

The trunk continues right to the top of the tree. This definition was developed primarily with conifers in mind (like the one in the top image on the left). They have a central axis that often extents to the very top of the tree, with all the branches coming off it in a similar way. It seems natural to call this the trunk. If you apply this definition to the banner image it should make a lot of sense. But if you apply it to the fractal tree you get strange results again – every time there is a split, one of the stems is called the trunk and the other is now a branch. That leads to the trunk taking a pretty random path to the top of the tree. That seems to also lead to a pretty crazy definition.

Now think back to the question we wanted to ask: how much of the mass is contained in the branches and how much in the trunk? Is the tree symmetric about the trunk? Clearly, the answers to these questions are going to depend on how we define the trunk and branching order.

Real trees

Shockingly, real trees don’t look like my neat idealized trees in the diagrams above! They are messy, with branches sticking out all over the place. The ‘real trees’ I am dealing with are actually cylinder model recreations of real trees, based on terrestrial laser scanning data. These are great but the underlying assumption, that trees look like a series of cylinders, breaks down at exactly the point we are interested in – the branching point. Before and after the branching point the tree is basically cylindrical, but at that point one branch is emerging out of the stem, leading to a strange shape. This causes the cylinder fitting algorithm we use to give errors, and to sometimes put multiple cylinders right next to each other, or mistake which branch is connected to which.

This is what the real trees look like, coloured by branching order according to the two definitions. In making these plots I have deleted lots of the smaller branches to make things clearer. Nevertheless, there are still lots of small cylinders, which represent branches, poking out all over the place. Each time this happens, according to definition A, the branching order increases. This leads to the many different branching orders, represented by many different colours, in the three trees in figure 5. Now this definition may give reproducible results but it seems pretty ridiculous, especially in the case of the pine tree on the right. If we want to know how much of the mass is contained in the trunk and how much in the branches, the trunk is just the turquoise bit down at the bottom, so is tiny compared to the rest of the tree.

The three trees in figure 6 are coloured by branching order as defined by definition B. this makes a lot more sense for the pine tree, and works OK for the tree in the middle too. However, you can see the problem with this method in the first tree in figure 6. The main stem splits in a large ‘Y’ shape quite low down. Since definition B requires the trunk (branching order 1) to continue to the top of the tree, the algorithm chooses either the right or left-hand path. This leads to an asymmetry throughout the tree. You could define a rule which chooses which path to follow based on a few factors (probably size, angle etc) but this would be sensitive to the quality of the cylinder fitting procedure and would require more arbitrary decisions, making the research less reproducible.

In summary, both definitions give some strange results, and this is partly due to the fact that the data is messy and our cylinder assumptions are not ideal. However, the fact that there are different possible definitions, and the results of some pretty basic questions depends strongly on which one we choose, is pretty worrying for me still! Sorry, maybe I should’ve warned you that this blog doesn’t have a solution. Next time you look at a tree, think about this problem and ask yourself – ‘are you confident you know which bit is the trunk?’

Finally, if you came here thinking about elephants and managed to make it right through to the end – have a look at this piece by Andy Burt.

References

Lau, Alvaro, Lisa Patrick Bentley, Christopher Martius, Harm Bartholomeusm, Alexander Shenkin, Pasi Raumonen, Yadvinder Malhi, Tobias Jackson, and Martin Herold. 2018. “Quantifying Branch Architecture of Tropical Trees Using Terrestrial LiDAR and 3D Modelling.” Trees.

Malhi, Yadvinder, Tobias Jackson, Lisa Patrick Bentley, Alvaro Lau, Alexander Shenkin, Martin Herold, Kim Calders, Harm Bartholomeus, and Mathias I. Disney. 2018. “New Perspectives on the Ecology of Tree Structure and Tree Communities through Terrestrial Laser Scanning. Interface Focus 8 (2). Royal Society: 20170052. doi:10.1098/rsfs.2017.0052.

da Vinci, Leonardo, Jean Paul Richter, and R. C. Bell. 1970. The Notebooks of Leonardo Da Vinci. New York: Dover publications.

Originally published at www.oxfordecosystems.weebly.com/blog