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Topic: Tree (the plant kind) Branching
Replies: 0

 David Boll Posts: 5 Registered: 12/12/04
Tree (the plant kind) Branching
Posted: Jun 14, 1996 4:19 PM

I was thinking about the way a tree branches, and wondering if there was
any mathematical relationship between the size of a branch before a fork,
and the sizes of the two branches after a fork.

Specifically, if a branch of size A forks into 2 branches of size B and C,
(size = diameter or circumference of the branch)
one is tempted to suppose that a relationship like:

A^x = B^x + c^x

will tend to hold for some value x. x=2 is clearly too small, 3 seems
about right - but I wasn't sure. I thought it would be cool if x turned out
to be equal to e, since e is the base of all logs. Har. Thought that one up
myself. No, really.

So, last night, under the disbelieving eyes of my neighbors, I went to
some trees in my back yard with a tape measure, and measured some branches.
(I measured the circumference in centimeters). Now, measuring the
circumference is not an exact science; care must be taken to stay close
to the fork (I tried to stay half a trunk-diameter from it), and also to
avoid any bumps, bulges, or out-of-round branches.

I only got 5 data points last night (which is 4 more than I need for an
exact theory :-) ), and here they are:

Trunk Branch1 Branch2
43.2 33.8 33.4
38.6 38 14.8
39 28 27
35 25.7 29
10.5 7.9 8.5

So, to test the theory, we can try to predice the size of the trunk given
the size of the 2 branches, using:

Trunk size = (B1^x + B2^x) ^ (1/x)

and then compare this theoretical trunk size to the actual one.

It turns out that x=2.82 predicts 4 of the 5 trunk sizes to within 1%

So - some more field research is definitely needed; I'll head out this
weekend for some more data points. Also - some theoretical justification
for x=2.82 would be nice. I suspect a case could be made for x=3, and
then various vigorous hand-waving arguments could be made for why x is a
bit under 3.

I also want to try trees of differend kinds (the above were all from
marine locust trees), and branches of larger sizes, to see if there is
some kind of universal tree branching constant.

---------------
Dave Boll dboll@hp-vcd.vcd.hp.com
"The speed of time is one second per second"