Tropical Forest Community Ecology

126 Egbert G. Leigh, Jr

increases disproportionately with the tree’s diam-
eter (Phillipset al. 2005). Finally, King ignored
reproduction, the central purpose of tree life.
Reproduction is costly: diameter ofTachigalitrees
increases four times faster than those of itero-
carpic canopy neighbors (Poorteret al. 2005).
Malaysian rainforest trees do not reproduce until
they are well enough lit to achieve a substantial
proportion of their annual reproductive potential
(Thomas 1996a,b,c), in accord with theoreti-
cal predictions of Iwasa and Cohen (1989). A
proper theory of tree height must incorporate
the trade-off between growth, reproduction, and
survival.

Trunk taper and tree shape

A tree’s shape reflects the trade-off between the
advantage of better-lit leaves and the costs of
supporting and supplying a taller crown (Givnish
1988). The first step to understanding this trade-
off is to learn what factors govern the design of
tree-trunks.
One criterion proposed for designing a canopy
tree’s trunk is that, when wind blows upon its
crown, th eproportional str etch (th estrain) on
the most stressed fiber is the same for all dis-
tanc es abov eth eground. If so, th ecub eD^3 (y)of
this trunk’s diameterymeters below the crown’s
center of mass is proportional toy(Dean and
Long 1986, Westet al. 1989). To show this, let
a forceFon the crown’s center of mass exert a
torqueyFabout a point on th etrunkymeters
below. Then the strain on the most stressed fiber
at levelywill b eproportional toyFD(y)/K(y),
which w eassum e equal to a constantcindepen-
dent ofy. Here,K(y)is th ecount ertorqu e excit ed
in the trunk by this stress. This countertorque is
ex ert ed by th ejoint action of th epull of th efib ers
on th estr etch ed sid eof th etrunk and th eir push
on the compressed side. To calculate this counter-
torque, consider paired fibers a distancexto either
side of the unstretched neutral plane that splits
the trunk longitudinally into stretched and com-
pressed halves. The countertorque from this pair
of fibers is proportional to the restoring forcekx
resulting from each fiber’s strain, times their lever
armxfrom the neutral plane. For the “average”
pair, this countertorque is proportional toD^2 (y).

Th etotal count ertorqu eis proportional toD^4 (y)

• the number of fiber-pairs involved, which is pro-
  portional to the trunk’s cross-sectional area, and
  thus toD^2 (y), times the average torque per fiber-
  pair, also proportional toD^2 (y)(see Leigh 1999,
  pp. 90, 113). IfyFD(y)/K(y)is proportional to
  yFD(y)/D^4 (y)=c,yis proportional toD^3 (y). The
  cube of trunk diameter increases linearly with
  distance below the crown’s center of mass until
  one reaches the butt swell for 45-year-old Douglas
  firs,Pseudotsuga menziesii, in western Washington
  (Longet al. 1981), matur elodg epol epin e,Pinus
  contorta, in northern Utah (Dean and Long 1986),
  and mountain-ash,Eucalyptus regnans, planted in
  Tasmania (Westetal. 1989).
  A rival criterion of tree-trunk design is that they
  have a fixed safety factor against buckling under
  their own weight (McMahon 1973, McMahon and
  Kronauer 1976). If so, 20% less wood is needed to
  support the crown with a given safety factor if
  D^2 (y), th esquar eof a trunk’s diam et erymeters
  below its crown’s center of mass, rather thanD(y)
  orD^4 (y), is proportional toy(King and Loucks
  1978, p. 149).D^2 (y) declines linearly with dis-
  tanc eup th etrunk for asp ens,Populustremuloides,
  in Wisconsin (King and Loucks 1978, p. 155),
  and, starting3mabovetheground, forSchefflera
  morototoniin Panama (tabl e5.5, p. 92 in L eigh
  1999).
  Nonetheless, our failure to progress beyond
  these crude models of support costs is a major
  obstacle to understanding tree shape. Tropical for-
  est has a great variety of “tree architectures”
  (Hallé and Oldeman 1970, Halléetal. 1978, Leigh
  1999). Our inability to predict their support costs
  is a primary reason why we usually cannot detect
  what advantage, if any, is peculiar to a given
  model.

Forest structure

Most natural forests have trees of all ages, diam-

eters, and heights up to the maximum. What

governs a forest’s distribution of tree diameters

and tree heights? Kohyamaet al. (2003) showed

how to relate a forest’s distribution of tree diame-

ters to tree death and growth rates. Let the number

N(D)of trees per hectare with diameters between

DandD+1 be constant for all integersD≥10 cm.