Life in the canopy: the tallest tree you'll ever see

Classical Mechanics Level pending

At this point, we've established the connection between the height of a tree and the range in the size of its leaves. At the upper range, leaf size stops increasing at the point of diminishing returns, where the gap between energy production and energy usage is at a maximum. At the lower range, leaves become so small that their sap flow loses out to molecular diffusion.

Our analysis has shown that the maximum leaf size is a strictly decreasing function of $$h$$, and that the minimum leaf size is a strictly increasing function of $$h$$.

If this is true, there must be a point $$h_c$$ where the min and max collide, i.e. where $$l_{min}(h_c) = l_{max}(h_c)$$.

But this also means that for all $$h > h_c$$, we have $$l_{min}(h) > l_{max}(h)$$, an impossibility!

We therefore have a very exciting prediction, that the maximum possible height for a tree is found at $$h_c$$. Above $$h_c$$ it would not be possible for the tree to construct a leaf capable of meeting the minimum energy flow rate while also being an economical investment for the tree to make.

Interestingly, below $$h_c$$, a tree's leaves can take any size in the range $$\{l_{min},l_{max}\}$$, and indeed, this is true. Smaller trees have a wide range of leaf sizes while the tallest trees have a very narrow window of potential leaf sizes.

The last question is, given the minimum leaf size (calculated in The Smallest Leaf), and maximum leaf size calculated in The Biggest Leaf, what is the value of $$h_c$$ (in m, the biggest possible tree height?

Hint: You may assume $$2\rho_{suc}E_{suc}L_p \gg \gamma_{leaf} r$$.

Bonus

• Compare this value to the tallest flowering tree in the world, Australia's Centurion, which stands at 99.6 m tall (picture above).
• This set took inspiration from many places, but the main calculation is an extended treatment of Jensen, and Zwieniecki's excellent PRL letter.
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