A RELATIONSHIP BETWEEN CIRCUMFERENCE AND WEIGHT IN TREES AND ITS BEARING ON BRANCHING ANGLES. (From Harvard University, Cambridge.

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1 A RELATIONSHIP BETWEEN CIRCUMFERENCE AND WEIGHT IN TREES AND ITS BEARING ON BRANCHING ANGLES. By CECIL D. MURRAY. (From Harvard University, Cambridge.) (Accepted or publication, February 21, 1927.) When a tree, at a point where the circumerence is Co, divides into two branches (cl and ct), what relationship exists between co and cl + c~? In order to answer this question, which has a deinite bearing on problems o tree orm, it is convenient to investigate irst the relationship between the circumerence at some point and the weight, w, o all the parts o the tree peripheral to this point. Accordingly measurements were made, 116 in all, on nine kinds o trees; namely, aspen, bitternut, hickory, oak, ash, maple, cedar, hornbeam, and beech. The largest tree measured had a circumerence o 56.4 cm. where cut, and the whole tree weighed 120 kg. The smallest measurements were made on the stems o leaves,--or example, circumerence o stem = 0.25 cm., weight o lea = 0.18 gm. All the data thus obtained are included in Fig. 1. Our procedure was o the simplest character. Whole trees o varying size, or branches, or leaves, were taken entirely at random rom the vicinity. The only criterion o selection was that the specimen should not appear to have been recently injured. The circumerence was measured by a tape, encircling the bark, at the point o section; or, or small specimens, the diameter was measured by calipers and the circumerence subsequently calculated. The specimen was then weighed on one o three balances according to size. The season was midsummer, 1926; the place, Grindstone Island, N. Y. Plotting logarithms; i.e., log (weight in gin.) vs. log (circumerence in cm.), the points all close to a straight line. A statistical treatment yields the ollowing numerical characteristics: Mean value o log c in the observations =- O. 161 " " " logw" " " ~ The Journal o General Physiology

2 726 RELATION BETWEEN CIRCUMFERENCE AND WEIGHT Standard deviation o log c = " " " log w Mean product o simultaneous deviations Correlation coeicient = 0.99 From these igures one obtains, or the best linear relation between log w and log c, the equation: 1.5. *Z/'* Y log w = 2.49 log c ; or, ~v ~= 7.08 c ~.,9 (1) j -g,. /. / / //.// 0 1 Z ~ 4 6 1o~e1~ht m ~) FIG. 1. The line is drawn according to equation (2). The trend toward a cube law relation among the observations appearing in the lower let portion o the chart may be signiicant. The probable error o log w is ,--i.e., i rom the measured circumerences estimates o the weight are calculated by equation (1), then hal o the actual observed weights all within the limits +20 per cent and -17 per cent o the calculated values. It will be seen rom Fig. 1 that the error is greater or the small pieces, and less

3 CZCIL V. mmaay 727 or the large pieces. We observed also that deviations rom equation (1) occurring in various parts o the same individual tree are suiciently large to mask any systematic dierences between the dierent kinds o trees that were studied. Furthermore, the equation holds as well or stems bearing nuts as or stems bearing leaves. Returning to the opening question, the solution, inherent in equation (1), is given by the relation: co s'(' i cl 1"(9 -I- c~ 1'*' (2)* This ollows rom the act that, i a main stem or trunk is cut near a point o branching and weighed (w0), and then i the branches are weighed separately (wl and w~), w0 must equal wl W w2. Equation (2) describes, or the class o trees studied, one special characteristic o branching. The exponent 2.49, being greater than 2, indicates, or example, that the total cross-sectional area o the branches becomes progressively greater at each branching. To express this property we may say that trees ollow statistically a "2.5 power law o branching." Another characteristic o branching is the equation which describes the angles o branching. In a previous paper 1 this problem was discussed in reerence to the arterial system in animals, and the ollowing equations, o which two only are independent, were deduced: co4 -[- c1( -- c~4 co 4 -- cl~ -~- c2 ( co ( -- ell -- c24 cos x i, cos y --, cos (x + y) i (3) 2 co 2 c1' 2 co scl ~ 2 cl ~ ct* where Co, cl, and c2 are the circumerences o the main stem and the two branches into which it divides; and where x and y are the angles made by the branches (c~ and c~) with the line o direction o the stem. The angle (x ~- y) is, o course, the angle included between the two branches. These equations, or our present purpose, may be considered as being deduced rom the assumption that the branching system connecting three points shall, or given circumerences o the stem and branches, require the least volume o wood. *Once obtained, this relation may be roughly checked by simple measurement, without weighing or cutting, on large trees. 1 The physiological principle o minimum work applied to the angle o branching o arteries, Murray, C. D., J. Gen. Physiol., , ix, 835.

4 728 RELATION BETWEEN CIRCUMFERENCE AND WEIGHT I now equations (2) and (3) are combined one can, at the expense o loss o generality, solve directly or the angle as a unction o some convenient ratio such as cl/co or cl/c2, The steps are shown in the previous paper. But in that paper, instead o a 2.5 power law (equation (2)), a cube law (theoretically deduced or the arterial system) was used. In either case certain qualitative rules hold which describe in words the variations in the angles accompanying variations in the ratio.cl/c~, etc. An interesting and convenient illustration o these / J 0g \ 3 5 EXPO NENT FIG. 2. The curve shows the relation between the angle (x + y), given that cl -- c2, calculated by equations (2) and (3), and the exponent, considered variable, occurring in equation (2). J rules may be seen in the branching o the veins o leaves. There remains only to be observed the act that, in changing rom a cube law to a 2.5 power law, the calculated angles, or any given ratio o circumerences, become smaller,--a act corresponding to a dierence between the branching o arteries and o trees. For example, solving equation (3) or the angle (x + y) when cl/c2 = 1, we ind or a cube law angle (x + y) = 75, or a 2.5 power law angle (x + y) = 59. The curve in Fig. 2 shows this relation.

5 C~.CIL D. ~P~AY 729 SUMMARY. Observation reveals a linear relationship bctween.the logarithm o the circumerence o a tree, branch, or lea stem, and the logarithm o the weight o the tree, branch, or lea. The bcaring o this on the angles o branching in trees is discussed.