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1.0
0.8
13
Koyama & Smith — The length-times-width equation for shoot leaf area
Magnolia kobus
Cardiocrinum cordatum
1000
4000
500
2000
500
Shoot leaf area (Ashoot) (cm2)
1000
Prunus sargentii
4000
6000
Ulmus davidiana var. japonica
600
1500
400
1000
200
500
12 000
2000
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1000
1500
100
200
300
400
500
Fallopia sachalinensis
9000
6000
3000
2500
5000
7500
(1/2) N (max + min single leaf area) (cm2)
Fig. 10. The total leaf area of a shoot (Ashoot) is proportional to the product of the number of leaves (N) times (maximum + minimum individual leaf area) divided
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APPENDIX
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15
16
Koyama & Smith — The length-times-width equation for shoot leaf area
shoot
shoot
By taking the product of both sides of the two lines in
eqn (14), and using eqn (2), we obtained:
mean (Aleaf ) ∝ max (Aleaf )
(15)
shoot
shoot
In deriving eqn (15), we assumed that the leaf that has maximum (or mean) length within each shoot also has the maximum (or mean) width in the same shoot. By multiplying both
sides of eqn (15) by the total number of leaves on each shoot
(N), we obtained:
N · mean (Aleaf ) ∝ N · max (Aleaf )
(16)
shoot
shoot
Because the left-hand-side of eqn (16) is Ashoot, we obtained
eqn (8).
Derivation of eqn (9)
In this section, we assume that lamina length and width of an
individual leaf are approximately proportional to each other
(Ogawa et al., 1995), and that individual leaf area can be predicted as the quadratic function of either leaf length or width
alone (Teobaldelli et al., 2019a, b):
Aleaf ∝ (Lleaf )
(17)
Equation (17) can also be used to predict the maximum Aleaf
within each shoot:
2
(18)
max (Aleaf ) ∝ max (Lleaf )
shoot
shoot
By multiplying both sides of eqn (18) by the total number of
leaves on each shoot (N), we obtained:
2
(19)
N · max (Aleaf ) ∝ N · max (Lleaf )
shoot
shoot
By combining eqns (8) and (19), we obtained:
2
(20)
Ashoot ∝ N · max (Lleaf )
By combining eqns (6) and (21), we obtained:
W f β+1 ∝ N · W f 2
(22)
Equation (22) can be solved for Wf:
W f ∝ N β−1
(23)
By substituting eqn (23) into eqn (6), we obtained:
β+1
(24)
Ashoot ∝ N β−1 ≡ N α
Derivation of eqn (10)
Equation (9) can be solved for Nf:
N ∝ Ashoot α
(25)
Using eqn (25), we obtained:
Ashoot
1− 1
∝ (Ashoot ) α ≡ (Ashoot )
(26)
Derivation of eqn (11)
By combining eqns (5) and (23), we obtained:
(27)
Lf ∝ W f β ∝ N β−1
By substituting eqn (27) into eqn (4), we obtained:
(28)
Ashoot ∝ W f · N β−1 ≡ W f · N γ
Derivation of eqn (13)
When different-sized foliage sets are affine to each other, the
maximum leaf size relative to the mean value and the minimum
leaf size relative to the mean value would be constants, independent of foliage size. This leads to the following relationship:
mean (Aleaf ) ∝ min (Aleaf )
(29)
shoot
shoot
By dividing both sides of eqns (15) and (29) by 2, and adding
them, we obtained:
shoot
By using our definition of Wf (eqn 3), eqn (20) can be rewritten as follows:
Ashoot ∝ N · W f 2
(21)
min (Aleaf ) + max (Aleaf )
shoot
(30)
mean (Aleaf ) ∝ shoot
shoot
By multiplying both sides of eqn (30) by N, we obtained
eqn (13).
Downloaded from https://academic.oup.com/aob/advance-article/doi/10.1093/aob/mcac043/6555708 by library@obihiro.ac.jp user on 12 May 2022
mean (Lleaf ) ∝ max (Lleaf )
shoot
shoot
(14)
mean (Wleaf ) ∝ max (Wleaf )
...
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