Tree Equations and Alien Plants
What physical laws govern the
structure, shape and form of trees?
Can we construct a mathematical model
and use it to predict the nature of alien
trees growing on other planets?

In this section we look at some key
equations describing plant/tree form
and function and consider what the
implications are for alien life.
Equation 1 is derived from Euler's column
equation and is for both solid and hollow
columns. The maximum allowable height of the
column before it buckles under its own weight
(ignoring additional stresses like wind forces)
is hcrit, EI is the flexural stiffness. Details are
given below. Equation one can actually be
simplified further, and this is done below when
we put it to use. E is the elastic modulus and I
is the second moment of area - not to be
confused with the second moment of inertia of
the mass, also called simply the moment of
inertia - moment of area is concerned with the
spatial geometry of an object in cross-section,
and gives us a measure of how spread out the
material, spatially, is from the central axis,
whereas the moment of inertia is concerned
specifically with the distribution of mass and
gives us a measure of how spread out the
mass is from the central axis. A wide cylinder
has a higher moment of area than a narrow
cylinder. Here we generalise to a hollow
cylinder with an outer radius, ro, and an inner

Equation 2 models a branch like a cantilever
(a horizontal beam fixed at one end and
loaded at the other) and Eq. 2a gives the
maximum moment (turning force due to
maximum deflection as the cantilever bends
under its own weight. Again this ignores
by being angled upwards.

More equations may be added to this list as
we develop the model. First of all let us look at
equation 1 and what it tells us.
The answer we obtained for the maximum or critical height, h, of 85 m is only approximate. We have estimated values
for the radii and mass and a real tree is tapered to some degree and more-or-less conical not cylindrical (we return to
this later). However, the answer is about what i expected - real trees have safety factors of 4 or more, which means
that a wood-cutter must remove at least 3/4 of the wood from a cross-section of the trunk before a tree begins to
collapse under its own weight (assuming negligible winds). Oak trees are typically 30 m in height, so this gives us a
safety factor of 3 - about right. A safety factor of 4,incidentally, is much higher than commonly used in
human-engineered structures (where the safety factor is about 2) - trees have to cope with the unexpected, such as
high winds, frost damage, grazing and lightning damage; trees are built to last!

Next, we make this critical height a function of gravity. On Earth the force due to gravity is about 10 Newtons (9,8 N).
How tall might trees grow on planets with more or less gravity than this?
Below we begin with Euler's column equation for an ideal column (an ideal column is one that is straight,
homogeneous and free from initial stress. This equation allows us to determine the critical load (P), or conversely the
maximum critical height (h) before a column permanently warps under its won weight - it assumes only self-loading
and so only looks at (vertical) compression stresses, and ignores, for example, wind forces which cause lateral
(sideways) forces to act on the column too. The elastic modulus is a measure of the stiffness of the material making
up the column, but the overall stiffness of the column also depends upon the geometry of the column, which is
included in the moment of area, I. In other words, we can estimate the maximum height of a tree before it collapses
under its own weight.
Note, some quoted variants on Euler's equation result in different coefficients, for example 1.95934 instead of 0.6168,
however, these equations are only estimates and we are more concerned with trends and will not worry too much
about the coefficient here. (Using 1.95934 gives a critical height of 103 m). Remember that trees strive to be tall in
order to compete for sunlight, however, there are limits, and from this result we might expect trees to be shorter on a
higher gravity planet, but not by that much - even when the gravity is 4o N (4 times stronger than on Earth) the
predicted height of trees is only halved.

Wood Density

Additionally, a tree could overcome this if it had less dense tissues. The effects of density on critical height are shown
below:
Tapered Trunks

A tree like balsa has a density as low as 380 kg/m^3 (critical h = 88 m), a redwood 450 kg/m^3 (critical height = 83 m),
whilst an ebony might have a density as high as 1120 kg/m^3 (critical height = 61 m). Now redwoods can reach heights of
over 115 m. This discrepancy can be explained partly by the fact that redwood trunks strongly taper, indeed all tree
trunks taper to a greater or lesser degree. A tapered stem reaches the same height by using less material (lower mass)
and also distributes stresses evenly throughout the column (in a cylindrical column, the stresses are greatest at the
base, remember that stress is force per unit area in this case). For a tapered trunk, the critical height increases. For a
trunk in which the topmost radius is half the basal radius (a truncated cone) the critical height increases by 26%, whilst a
greater taper may increase it by up to about a third or more. (The coefficient is increased).

Solid Trunks

The results for a solid cylindrical tree trunk are shown below:
At am Earth-like gravitational force of 10 N, being hollow for our tree (outer radius 0.5 m, inner radius 0.4 m) increased
the critical height by 18%. However, for a larger tree (outer radius 1.0 m, inner radius 0.9 m) the increase was 22%,
whilst for a narrow tree with outer radius 0.25 m and inner radius 0.15 m the difference is only 11%. Thus, for a constant
wall thickness, larger trees derive more benefit from being hollow. (If we scale the wall thickness proportionally to
increase in trunk diameter then there is no difference).

Being hollow is much more importance when we consider lateral wind forces. Increasing trunk diameter increases the
moment of area (I) and so increases flexural stiffness. Being hollow means that a trunk of given mass can have a larger
diameter - being hollow increases trunk stiffness whilst economising on material. Experiments have shown that high
winds are more likely to topple the tall, solid and thin younger trees than it is to topple the older, larger but more hollow
trees. Hollow trees may also allow wind to pass through them more easily. The alien tree in the picture at the top of the
page is adapted for high winds - the wind easily passes through the spaces in its trunk and through the slits in its
dome-like crown.

Stem Shape

Some plants, especially small green herbaceous plants, have square stems. For a given amount of material (constant
cross-sectional area) a square stem has a slightly higher moment of area and so is stiffer. This is an advantage for
these softer green stems. Many trees have elliptical stems and branches. This is because the tree responds to the
stresses acting upon it as it grows. In an ellipse one radius (the major radius) is larger than the other (minor) radius and
in these plant parts the long axis will be aligned with the direction that experiences greatest bending stresses -
increasing moment of area and stiffness whilst economising on the total mass of material used.
Branches can be modelled using a modification of the column equation for a beam. The results are similar and as gravity
increases the branches are predicted to shorten in proportion to the shortening of the stem, so that the overall tree
shape is maintained.
Water Transport

So far we have only considered the effects of gravity on tree shape. Also of importance is the need for a tree to move
water up the trunk from the soil, via the roots, into the leaves. The function of the leaves is to obtain sunlight and carbon
dioxide for photosynthesis. To achieve this, the leaves must be thin and plate-like (otherwise too many
non-photosynthesising cells would occupy the centre of the leaf where light cannot penetrate). They also need leaf pores
(
stomata) to let the carbon dioxide gas pass from the atmosphere into the leaf. A consequence of this is that water will
also escape by evaporation from these pores (a process called transpiration) and this water must be replaced by water
moving up the plant from the roots in what is called the transpiration stream. However, plants exploit the transpiration
stream to carry mineral nutrients from the soil to the cells and transpiration also helps the sunbaked leaves to keep cool.

Transport in plants
Equation 3 - The Transpiration equation for Stomatal Conductance

Using Fick's Law we can derive an equation to model diffusion across the stomata. Essentially, this is one-dimensional
diffusion across stomata modeled as narrow tubes. However, the diameter of the tube is also important and a number of
variants on this equation incorporate corrections to account for this. We use one such simple correction below. Note that
the variants given of this equation all give values of conductance within an order of magnitude of one-another and the
patterns and trends are essentially the same (all the ones I have seen differ only by a constant of proportionality).

Plants can open and close their stomata in response to a variety of internal and external conditions and also in
accordance to an internal clock. In addition, in the long term, plants can vary both the number (density of stomata per
square mm of leaf surface area) and the maximum size of the stomatal apertures. These characteristics differ greatly
between species, but are also capable of some adaptive variation within a species. For example, it has been shown that
as carbon dioxide levels have increased, both by natural means since the last peak glacial period of the Ice Age and
more recently from anthropogenic emissions, plants respond by  reducing stomatal conductance. This is achieved by
having fewer, larger pores per leaf, as explained below:
Equation 5 - The Growth Equation

A tree must invest a lot of resources in its stem, to maintain the dominant position of its leaves high in the canopy. The
trunk of large trees is not photosynthetic at all and so these materials do not bring direct gain in terms of growth. The
growth of a tree is driven by its leaves (supplemented by nutrients and water from the roots). The fastest growing plants
are almost all leaf. Duckweed is one of the fastest growing of all plants, and each plant consists of a little leaf and a tiny
root and the plant floats in the water. Single-celled algae (protoplants) are even faster growing. In contrast, a large tree
which must invest so many materials in supporting structures is very slow growing relative to its size, that is in terms of
percentage  weight or mass increase.

The growth equation is:
This gives us the growth rate, R, which is the rate of increase in mass relative to the present mass. This is relative
growth (and obtained by logarithmic differentiation).

A young sapling grows more-or-less exponentially and relatively very fast. However, once a tree reaches its full height,
the trunk continues to grow by adding annual rings of new wood to the outside of the trunk, beneath the bark. Typically
the cross-sectional surface area (and hence volume and mass as height is now constant) of new wood added each year
remains approximately constant for a mature tree. (It is well known that it varies from year to year according to
environment, but the average rate is more-or-less fixed). In the end, their is not enough new wood added to encompass
the trunk and parts of the tree start to die back and eventually the tree enters decline.

Thus, for a mature tree, the rate of increase in mass per unit time (dM/dt) is essentially constant and the growth equation
becomes:
The plot shows the growth rate (this time given the symbol G) as a function of mass. This relative growth declines, as the
yearly addition of mass represents a diminishing fraction of the tree's mass. (The units of mass in this plot are arbitrary).

Actual growth curves have been obtained for trees by taking measurements. For the yew tree, one of the longest tree
species (of which a number of specimens are dated to around 3000 to 5000 years of age) the following was obtained (by
measuring trees up to 1000 years of age):
This matches our picture of slowing relative growth, though in the case of the yew this growth can
continue for a remarkably long period of time!
This effect is also reflected in our conductance equation, shown below, in which conductance, G, is not only a
function of stomatal depth, d, and total leaf area accounted for by stomatal pores (stomatal density, n, multiplied by
the mean pore area of the stomata, a) but also of stomatal radius, r. If the total pore area is kept constant, then
stomatal conductance decreases as pore radius increases, as shown below:
Of course, if the stomata close, then both the pore radius and the total area of pores reduces and conductance
naturally decreases, as shown below. Notice from the graph that the stomata affect the greatest relative changes on
conductance at very small pore radius. Most stomata are between 3 and 15 micrometres (0.003 and 0.015 mm) in
radius and so can rapidly regulate conductance by slight changes in diameter:
Each species occupies a narrow part of such a conductance curve, with some species having intrinsically high
conductance (those from carbon dioxide poor atmospheres in which water is plentiful) whilst some have lower
conductance (those from carbon dioxide rich atmospheres, or regions of water shortage). The fact the increasing the
levels of carbon dioxide causes plants to adapt by reducing conductance (whilst still possibly increasing net
photosynthesis) reduces stomatal conductance, shows the importance of conserving water - stomata function primarily
to allow carbon dioxide to diffuse into the leaf, and although the transpiration stream in the xylem transports some
useful materials, like minerals from the roots, this function is secondary and xylem transport is in excess of that
required for these transport functions, serving primarily to replenish water lost by the leaves through their stomata.
However, i am not aware of any studies quantifying the importance of mineral transport - could plants without stomata
obtain sufficient nutrients without flow in the xylem?

Within a species, maximum pore area and stomatal density typically vary by 2-fold in response to environmental
conditions. Densities may vary from 100 to 1500 per square mm in different species, and maximum pore areas from 30
to 170 square micrometres.
The graphs below show the shapes of curves of stomatal area as a function of stomatal density (now given the symbol x so
as not to confuse Mathcad, x = n) different conductance values. See how at higher conductances the curve shifts up and to
the right. Similar plots can be seen in many papers on plant physiology.
What about alien plants? We might expect those plants that live in a carbon dioxide rich atmosphere to have fewer larger
pores for carbon dioxide absorption. Carbon dioxide may also, however, have major effects on tree branching patterns
and leaf shape, as we shall explore qualitatively below (the diffusion equation).
Equation 4 - Transport Equation (Poiseuille's Law)

One very important equation for plant transport is the equation for water potential (see transport in plants). Here we look
at the equation governing flow in xylem and phloem: Poiseuille's Equation for laminar, parabolic flow in a straight tube:
See transport in plants for a more detailed description of the physiology of sap transport in plants, a brief summary of
which follows below.

This flow describes flow in xylem especially, but also bulk flow in phloem, quite accurately. However, in phloem, different
forces give rise to the pressure gradient which drives the fluid flow. In xylem, the pressure gradient is a negative suction
driven by evapotranspiration of water from the tree canopy. Evapotranspiration consists of evaporation through the
cuticle of the leaves and also transpiration through the open pores or stomata of the leaves which take in carbon dioxide
for photosynthesis. The need for carbon dioxide makes this water loss largely unavoidable and sets up a pressure
gradient that sucks water up the stem through the xylem conduits. This stream of xylem sap is used by plants to carry
mineral salts as nutrients from the roots to the aerial parts. There is also a small positive pressure contribution from the
roots, called root pressure, which results from the active (energy-consuming) pumping of salts by root cells from the soil
into the xylem. This pumping is necessary because of the relatively low soluble mineral content of soil and draws water
after it by osmosis.

In phloem, generally positive pressures drive phloem sap through the phloem conduits from one part of the plant to
another. Generally organic building-blocks and energy sources manufactured in the leaves and other green aerial parts
are transported in the phloem to where they are needed, but phloem can flow in any direction up or down a plant. The
transport of sugars into the phloem at
sources and its removal at sinks drives water with it by osmosis and creates the
pressure needed for bulk phloem flow.  Sources include photosynthesising leaves, storage organs which are mobilising
their food reserves, e.g. bulbs in early spring, whilst sinks include growing plant parts, roots, heavily shaded leaves,
growing storage organs (such as a bulb filling with food reserves before winter) and developing fruit. Flow in the phloem
also has a pulsatile quality, which is thought to be due to the loading of sugars into the phloem at regular intervals by
companion cells.

Note that the flow rate of sap is proportional to the square of vessel diameter. The cross-sections of xylem vessels varies
considerably. Broad-leaved trees, like oak trees, produce wide-diameter vessels in new wood (early or Spring wood)
deposited in Spring and narrower vessels in summer. The early vessels enable the rapid transport of xylem sap for rapid
spring growth and to replace the water lost by evapotranspiration when the new foliage opens. However, late or summer
wood has smaller diameter vessels, since the high pressures generated by evapotranspiration in the summer heat is
more likely to cavitate larger diameter vessels. Cavitation occurs when the water column breaks and air fills the space,
resulting in a bubble which is difficult to shift and which may block a vessel temporarily or even permanently. Cold
conditions also increase the likelihood of cavitation and so cold-hardy conifers have much narrower vessels (tracheids).

Each xylem vessel may be several centimetres to a meter or so in length at which point the sap moves across into
another adjoining vessel, such that the xylem sap takes a somewhat zig-zag path up the stem. This is beneficial, since it
allows sap to circumvent blockages or breaks due to wounds, etc. without depriving whole sectors of the crown.

In tiny plants, namely mosses, water moves up the plant by capillary action, whether it is moving up the outside of the
stem or inside the stem in more-or-less specialised tissue. No other driving force is thought to be necessary to move
water up a plant only a few millimetres or centimetres tall, though evaporation likely also contributes.
Equation 6 - The Diffusion Equation and Branching Patterns

The diffusion equation is important in modeling the diffusion of carbon dioxide in to leaves where it can be utilised for
photosynthesis. The equation is given below:
For many situations we need to use computers to solve this equation by numerical approximation. (We have done this for a
simple 1D case in our article on
diffusion). This requires certain specialist techniques. An interesting study carried out on
sponges used the diffusion equation to model the delivery of food to a sponge. Sponges are animals, but they often
assume tree-like growth forms. They draw currents of water through their bodies and filter particles of food, such as
bacteria, from these currents. In still waters, diffusion (and also sedimentation due to gravity) must replenish the food
supply (clearly the circulating currents generated by the sponge assist but for simplicity let us consider only diffusion). The
water is drawn in through pores scattered over the body surface. Areas of the sponge compete with one-another for food -
one area may deplete the local region of water of food which an adjacent area needs. By growing in response to food
availability (and perhaps a genetic program) the sponge avoids this wasteful competition by branching in such a way that
each branch is supplied by the water around it (which in turn is supplied by diffusion from the water column). Branching
ensures maximum utilisation of the nutrients without producing tissues in regions depleted of food.

In the case of green plants, the leaves have to absorb carbon dioxide for photosynthesis. The final branching pattern must
ensure that leaves do not overlap their regions of supply too much, or else leaf tissue is surplus to requirements and
investing in tissues is expensive, especially if those tissues consume more than they produce. There are a couple of dozen
or so models of tree branching. Such models can be predicted from theoretical considerations, such as the need to obtain
carbon dioxide, and trees will generally fit one of these models, though not all models may have real-life examples.
Complications arise because a tree also needs to consider the optimum interception of light by the leaves, the amount of
material used in branch and twig construction and mechanical strength. These patterns are sometimes quite precise, but
often local environmental conditions will modify the pattern, as a branch responds to local light levels (e.g. shading by a
neighbouring plant) mechanical stresses, e.g. due to wind, and damage, e.g. by grazing animals.
information on tree branching patterns.

Branching is not just confined to the woody parts of trees. Leaves may themselves become sub-divided into leaflets or
lobes. Much of this dividing up of leaves is probably again governed by the need to acquire carbon dioxide, though in some
cases temperature regulation of the leaf may also be a factor: divided leaves may have better air-circulation for cooling. It is
worth recalling that leaves are not generally rigidly attached, but are attached via a flexible hinge called the
pulvinus. This
allows leaves to flitter in the breeze, especially in such plants like aspen, in which the slightest breeze will flutter the leaves
with a very audible sound. This may serve to mix the air, bringing in more carbon dioxide. In static air, a zone of carbon
dioxide depletion will accumulate around the leaves (the still boundary layer) and this would then be replenished by the slow
process of diffusion only. Stirring the air may serve tor educe the thickness of this still boundary layer. Mechanical factors
also need to be taken into account. A flexible leaf offers less resistance to wind, reducing loading on the branch and stem
of the tree. Larger leaves are especially divided, such as the compound leaves of horse chestnut (
Aesculus
hippocastanum
) and this again reduces resistance to wind.

A more dissected leaf, by reducing the boundary layer thickness, also increases cooling by advection and evaporation and
increases access to carbon dioxide and, probably more importantly, increases the efficient use of carbon dioxide as
explained above. The submerged leaves of aquatic plants are often very divided and feathery and this may correlate to the
slower diffusion of carbon dioxide in water, making carbon dioxide more limiting under water. Models with the diffusion
equation ought to be able to test this idea. Many plants have basal shade leaves with entire (undivided) margins whilst at
the top of the stem, the leaves are more divided and called 'sun leaves' and their may be transitional leaves in between.
Some have considered this a cooling mechanism to protect upper leaves from overheating in sunlight. However, sun leaves
presumably undergo more photosynthesis and so utilise more carbon dioxide, in which case a divided morphology is to be
expected. Again, models using the diffusion equation could examine these issues.

Dehydration is another factor affecting leaf morphology. Conifers often thrive in cold habitats in which leaves may suffer ice
damage and also in which the plants may experience water stress since much of the water may be locked up as ice: icy
environments are dry environments (xerophytic). Thus, a pine tree has narrow leaves which offer less surface area to
reduce water loss by evapotranspiration, covered with thick cuticles to again reduce water loss. These needles also shed