LAWS OF TALL-TREE GROWTH INVESTIGATED 



MATHEMATICALLY 



By Prof. R. D. Bohannan 



Ohio State University 



It is assumed in what follows that the trees are growing in suitable 

 localities, under forest conditions, with mutual protection from the 

 wind and sufficient crowding to destroy lateral buds, leaving the full 

 energy of the tree to the terminal bud, so that the tree, in its contest 

 for light with its neighbors, takes the form of a tall cone tapering under 

 a small constant angle to a small top. 



{A) Neglecting the influence of the top on the stability of the tree 

 and assimiing still air, the greatest height the tree can reach, without 

 bending under its own weight, if slightly tilted out of the vertical, is 

 given by the formulas: 



3-"^= 1.97!/^ , for cones. 



3— :i^ = 1 . 25 .*/_ , for cylinders (palms) . 



And if we assume the bole of the tree has the form of a paraboloid 

 of revolution (a supposition justified by experiments), the correspond- 

 ing formula is : 



It thus appears that of these three forms the cone-form allows the 

 greatest possible still-air height, and for this reason we have selected 

 this form for the determination of H. Selection of the paraboloid- 

 form would diminish our value of H by about 10 per cent and thus 

 increase the value of k in the equations of the form, h = kH, by about 

 15 per cent, but would not otherwise affect the conclusions. 



In these formulas r is the stump-radius in inches; E is the modulus 

 of elasticity in pounds per square inch for green wood of the species 

 being considered, w is the corresponding weight per cubic foot of the 

 same green wood, and H the height in feet. 



These formulas will be proved at the end of this paper. The value 

 of H given by the cone formula for any value of r will be referred to in 

 what follows as the greatest possible still air-height of the tree for that value 

 of r, and will be denoted by H. 



532 



