1. Allometric relationships between large and small scale tree spacing studies Mike Strub and Ralph Amateis

2. Topics Spacing studies Scaling – isomorphic versus allometric Comparison of small and large scale spacing studies Conclusions

3. Spacing Studies Full scale studies were planted at four locations with three replications in the spring of 1983. Treatments include 4’X4’, 6’X6’, 8’X8’, 12’X12’ and various rectangular spacings. A single replication of a 1/16th scale study was planted in the spring of 1989 and a second replication was planted in 1998.

4. Scaling Early work dates back to the development of geometry and notions about similar triangles, these ideas involve a single scaling factor and are termed isometric. Latter work came from using scale models of engineering projects and resulted in the development of dimensional analysis to address more complex allometric problems.

5. Scaling The dimensional analysis model is : Y=  *X^  These ideas have been applied to biological organisms in the last thirty years.

6. Scaling

7. Buckling Limit A tree will buckle under its own weight when: Which implies that to prevent buckling:

8. Buckling Limit from McMahon and Bonner (1983)

11. Buckling Limit from Horn (2000)

12. 4’X4’ spacing

13. 6’X6’ spacing

14. 8’X8’ spacing

15. 12’X12’ spacing

16. 4’X4’ spacing

17. 6’X6’ spacing

18. 8’X8’ spacing

19. 12’X12’ spacing

20. 4’X4’ tree spacing

21. 6’X6’ tree spacing

22. 8’X8’ tree spacing

23. 12’X12’ tree spacing

24. D=  *(H-4.5)^ 

25. full scale spacingalphabeta 40.6680.521 60.7950.531 80.8100.575 120.6940.679 miniature scale spacingalphabeta 40.7410.550 60.6090.664 80.5820.705 120.8680.644

26. Results Both miniature and full scale studies approach the buckling line in a similar fashion. Miniature scale studies have relatively larger diameters than full scale studies, likely because they were measured at ground line rather than scaled breast height. The miniature scale studies tend to be more variable than the full scale studies, perhaps because they are the results of a few years of weather rather than decades.

27. 4’X4’ tree spacing

28. 6’X6’ tree spacing

29. 8’X8’ tree spacing

30. 12’X12’ tree spacing