LAWS OF TALL-TREE GROWTH 533 



(B) Throughout long periods of the steady growth of trees the 

 following laws of growth hold: 



(1) The height divided by the cube root of the square of the stump- 

 radius remains nearly constant, this constant being different for differ- 

 ent species and dependent upon the energy of the terminal bud. 



(2) The height, throughout the same period, on any stump-radius, 

 maintains a nearly constant ratio to the greatest still-air height pos- 

 sible on the same radius. 



(3) The ratios mentioned in (1) and (2) generally increase rapidly 

 in youth until they reach their normal value (exceptions in opposite 

 directions are redwoods and Douglas firs; see special tables following). 

 These ratios diminish in old age. They are also affected by local en- 

 vironment, so that the tallest tree of any species need not be also the 

 tree of maximtun diameter. 



If h denote the actual height of the tree on the sttimp-radius r, laws 

 (1), (2) are equivalent to: 



/^=cV^ (a) 



h' 'jZ (6) 



ifH^) 



(d) 

 (e) 



Here h' is the height on the stump-radius r'; h" the height on the 

 stump-radius r"\ H', H' the greatest possible still-air heights on r', r" . 

 Equation (c) will, of course, show greater divergence than (6), since a 

 given divergence in a cube root will show up threefold in the cube. 



(C) Since by {B) h=cr 



dh_^rydr^_c \ _^yjc^ 1 

 •• dt- dt~3 Vr~ ^Vf 

 That is, the time-rate of height growth divided by the correspond- 

 ing time-rate of stump-diameter growth, during long periods of the 

 steady growth of the tree, varies inversely as the cube root of the stump- 

 radius, or inversely as the square root of the height. Thus the bigger 

 and the higher a tree grows the more slowly will it grow in height for a 

 given rate of stump-growth. 



