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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 trunkdiameter from it), and also to avoid any bumps, bulges, or outofround 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% (the middle one is off by about 10%). Not bad!
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 handwaving 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@hpvcd.vcd.hp.com "The speed of time is one second per second"



