Tree forms
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Tree form Tree form theories Form ratios Taper equation
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Tree forms
- 2. •Tree form •Tree form theories •Form ratios •Taper equation •Tree form
- 3. • Tree bole tapers generally from base to tip dia with ht • Taper varies rate-wise & shape-wise for different species • Taper also get influenced by a wide range of environmental and contextual factors • Different growth rate for different parts of the bole • A complex interaction between the bole form & the tree crown, bole gets affected by the change in crown Tree Form
- 4. Tree Form Shape & branching habit vary with species Tree form is the shape of bole i.e. rate of diminution of diameter with height (base to tip) Required to know important aspects of form that may affect marketability
- 5. Stem or bole form • rate of taper of a log or stem • development of the form of stem depends on the mechanical stresses to which the tree is subjected • stress come from dead weight of stem and crown and the wind force • wind force causes the tree, to construct the stem in such a way that the relative resistance to shear is same at all the points on the longitudinal axis of the stem
- 6. • Complex, not easy to approximate in known geometric shape in total • Some general [geometric shapes] approximate portions of the tree bole, but there are many inflections and points of irregularity • Knowledge of the tree form can help in better estimation of bole volume or biomass better understanding of the growth conditions Tree Form
- 7. - straight bole - fine branches - no apparent defects etc. Perfect tree form
- 8. - not ideal - some kinks in stem - evidence of insect attack etc. Acceptable tree form
- 9. - crooked bole - severe butt sweep - forked - evidence of diseases e.g. rot Unacceptable tree form
- 11. Parts of a tree stem tend to approximate truncated parts of these common shapes. - base tends to be neiloid - tip tends to be conoid - main part of the bole tends to be paraboloid The points of inflection between these shapes, however, are not constant.
- 15. Tree form
- 16. The taper of a geometric shape where x = distance from the apex of the shape & y = dia Specific values of b correspond to common shapes 1. Paraboloid 2. Conoid 3. Neiloid y2 = k * x y2 = k * x2 or y = k` * x y2 = k * x3 What about cylinder?
- 17. Paraboloid: further divided into 1. Quadratic paraboloid where b = 1 y2 = k x i.e. curve of ht vs. (dia)2 is a straight line 2. Cubic paraboloid where b=0.66 y2 = k x0.66 or y3 = k` x i.e. curve of ht vs. (dia)3 is a straight line y2 = k * xb
- 18. •Tree form •Tree form theories •Form ratios •Taper equation •Tree form theories
- 19. Tree form - theories Nutritional theory Water conducting theory Hormonal theory Mechanistic theory or Metzger's beam theory
- 20. Nutritional theory and Water conducting theory are based on ideas that deal with the movement of liquids through pipes. They relate tree bole shape to the need of the tree to transport water or nutrients within the tree The hormonal theory envisages that growth substances, originating in the crown, are distributed around and down the bole to control the activity of the cambium. These substances would reduce or enhance radial growth at specific locations on the bole and thus affect bole shape.
- 21. Metzger’s Theory • Has received greatest acceptance so far • Tree stem - a beam of uniform resistance to bending, anchored at the base and functioning as a lever as a Cantilever beam of uniform size against the bending force of the wind • Maximum pressure is on the base so the tree reinforces towards the base and more material deposited at lower ends
- 23. • A horizontal force will exert a strain on the beam that increases toward the point of anchorage, and if the beam is composed of homogeneous material the most economical shape would be a beam of uniform taper Metzger’s Theory contd… Tree in open – short, with rapidly tapering boles Tree in close canopy – long & nearly cylindrical boles
- 24. Metzger’s Theory contd… P = force applied at the free end L = distance of a given cross section from the point of application of this force d = diameter of the beam at this point Then by simple rule of mechanics, the bending stress (kg/cm2 ) S = 32PL/π d3 d L P
- 25. Metzger demonstrated that this taper approximates the dimensions of a truncated cubic paraboloid (ht against dia3 is linear) after confirming his theory for many stems, particularly conifers. Metzger’s Theory contd… If W = wind pressure per unit area A = crown area Then total pressure P = W x A S = 32PL/π d3 ⇒ S = (32/π ) W A L /d3 For a given tree ⇒ d3 = (32/π ) W AL/S ≈ k L ⇒ d3 ≈ k L (Cubic paraboloid)
- 26. Metzger theory • Tree bole similar to a cubic paraboloid d3 ≈ k L (Cubic paraboloid) • Stem a beam of uniform resistance to bending anchored at the base, and functioning as a lever arm
- 27. • Wind pressure, acting on crown is conveyed to the lower part of stem, in increasing measure with the length of bole • Greatest pressure is exerted at the base of tree danger of the tree snapping at base Metzger’s Theory contd…
- 28. • Tree reinforces itself towards the base to counteract this • The limited growth material is so distributed that it affords uniform resistance all along its length to that pressure Metzger’s Theory contd…
- 29. • Trees growing in complete isolation have larger crowns so the pressure exerted on them is the greatest • If such tree is to survive, it should allocate most of the growth material towards the base, even though it may have to be done at the expense of height Metzger’s Theory contd…
- 30. Trees growing in dense forest are subjected to lesser wind pressure longer & cylindrical bole Metzger’s Theory contd…
- 31. Methods of studying tree form • By comparison of Standard Form ratios • By classification of form on the basis of form ratios • By compilation of taper tables
- 32. •Tree form •Tree form theories •Form ratios •Taper equation •Form ratios
- 33. Form Ratios – form factor & form quotient Form Factor summary of the overall stem shape ratio of its volume to the volume of a specified geometric solid of similar basal diameter and height Most commonly, the form factor of trees is based on a cylinder form factor = vol. (stem) /vol. (cylinder) or tree volume = form factor x basal area x height
- 34. Form Factor • The standard geometric shape for the bh form factor is a cylinder of the same height as the stem and with a sectional area equal to the sectional area of the stem at bh (i.e. basal area) • Form factor is the ratio of the volume of the stem to the volume of the cylinder
- 35. Classification of Form Factor Depending upon the height of measurement of basal area Artificial form factor – vol. (whole tree) / vol. (cylinder with basal area at bh) Absolute form factor – vol. (tree above point of measurement) / vol. (cylinder at same level) Normal form factor – vol. (whole tree) / vol. (cylinder with measurement at some % of height)
- 36. Artificial form factor or Breast height form factor – the most common • Standard geometric shape for this form factor is a cylinder • Height of cylinder is same as the height of stem • Sectional area of cylinder is equal to the sectional area of the stem at bh i.e. basal area F = V/S x h F = form factor V = volume of tree S = basal area (at bh) h = height of the tree
- 37. Absolute form factor Defined as the ratio of the volume of the tree above point of measurement and the volume of a cylinder (of same dia or basal area) at same level Not used
- 38. Normal form factor True form factor Defined as the ratio of volume of whole tree and the volume of a cylinder with measurement at some % of height (generally 1/10th , 1/20th …. of total height) Not very convenient, requires prior measurement of the height of tree before deciding the points of measurement Not much used
- 39. Specific breast height form factors Cylinder 1.00 (>0.9) Neiloid 0.25 (0.2-0.3) Conoid 0.33 (0.3-0.45) Quadratic paraboloid 0.50 (0.45-0.55) Cubic paraboloid 0.60 (0.55-0.65) If the appropriate bh form factor for a tree of a given age, species and site can be determined, then the stem volume is easily calculated by multiplying the form factor by the tree height and basal area.
- 40. Form quotient • Ratio of the diameter at two different places on the tree • Generally calculated for some point above bh to the dbh • Absolute form quotient – most common dbh bh
- 41. Absolute form quotient • Grouped into form classes • Calculated by measuring the dia at a height halfway between bh and total tree height • Absolute form quotient = dia at halfway/ dbh • Commonly written as d5/d0 and expressed as a decimal e.g. 0.70 Form quotient contd…
- 42. Form class Defined as one of the intervals in which the range of form quotients of trees is divided for classification and use Also applies to the class of trees which fall into such an interval Form quotient interval such as 0.50 to 0.55, 0.55 to 0.60 … or mid-points of these intervals e.g. 0.525, 0.575 …
- 43. Absolute form quotients also suggest general stem shapes: Neiloid 0.325 - 0.375 (FQ class 35) Conoid 0.475 - 0.525 (FQ class 50) Quadratic paraboloid 0.675 - 0.725 (FQ class 70) Cubic paraboloid 0.775 - 0.825 (FQ class 80) Form class & quotient contd…
- 44. Form Point - Located approximately at the centre of gravity of crown since crown offers max. resistance. This point is the focal point of wind force. Form Point Ratio - Percentage ratio of the height of the Form point to the total tree height. - The greater the Form Point Ratio the more cylindrical tree - However, this point is difficult to locate in crown (Subjective)
- 45. Tree taper Defined as the change in stem dia between two measurement points divided by the length of the stem between these two points Taper equations attempt to describe it as a function of tree variables such as dbh, height, etc.
- 46. •Tree form •Tree form theories •Form ratios •Taper equation•Taper equation
- 47. Taper equations Try to fit the stem profile in model form and predict the dia at any point on the tree stem Based on simple input variables like dbh and total tree height, are unbiased Numerous approaches for the construction of these equations such as: Use of computers to fit complex polynomial equations Often the tree bole was segmented into 2 or 3 parts and separate equations fitted to each part Development of a model that describes the continuous change in stem form from ground to tip
- 48. Efforts have been made to discover a single, simple two- variable function involving only a few parameters which could be used to specify the entire tree profile but limitations are due to infinite variety of shapes of trees and other physical features Taper equations contd…
- 49. Taper equations contd… Can be used to provide predictions of inside bark dia at any point on the stem estimates of total stem volume estimates of merchantable volume and merchantable height to any top dia and from any stump height estimates of individual log volumes
- 50. Taper Table Taper Table portrays stem form in such a way that the data can be used in calculation of stem volume actual form is expressed by dia at fixed points from base to the tip of the tree if sufficient diameters are taken at successive pts. along stem, taper tables can be prepared
- 51. Diameter Taper Table : gives taper directly for dbh without referring to the tree form Form Class Taper Table : Dia at different fixed points on the stem expressed as % of dbh (ub) for different form classes
- 52. Equations of tree form Despite the fact that trees don’t conform to any geometric shape, some equations have been derived to describe tree form Diameter quotient = (d at any given point)/ dbh Behre’s Formula d/d.b.h. = l / (a+b * l) a & b are constants such that a+b = 1 l is the distance from the tip of the tree (% of length of tree between bh & tip)
- 53. Hojer’s Formula d/dbh = C log [(c+l)/c] d is Dia at ‘l’ (same as previous) C & c are constants for each form class “Trees assume infinite variety of shape”-- explicit analytic definition of tree form requires considerable computational effort -- yet lacks generality”. Hence simple functions, graphical methods is adequate for most purposes.
Hinweis der Redaktion
- The diameter of a tree bole generally decreases or tapers from the base to the tip. The way in which this decrease occurs defines the bole form. This taper can occur at different rates and in different ways or shapes. An understanding of the form of the bole can allow: Improved estimation of bole volume or biomass (when used in conjunction with dbh) Improved estimation of the presence and amount of wood products (by product specifications) Improved understanding of the competition and growth conditions of the tree There is a complex interaction between the bole form and the tree crown. Thus, any factor that influences the crown may also influence the bole form. Different parts of the bole grow at different rates as environmental and other factors affect the crown and the way photosynthesis are distributed.
- The shape or branching habit of a tree can affect its commercial value markedly. When assessing trees it is useful to record any important aspects of form that may affect marketability. The perfect "target tree" for saw milling might, for example, have a very straight butt log with a single leading stem. The shape or branching habit of a tree can affect its commercial value markedly. When assessing trees it is useful to record any important aspects of form that may affect marketability. The perfect "target tree" for saw milling might, for example, have a very straight butt log with a single leading stem. Although most field recording sheets do allow for comments against each tree, it is helpful to be able to more quickly record a summary of each tree's form and suitability for the intended use or market. There are many different methods for doing this. As a minimum it is recommend that when measuring trees farmers classify each tree as having either: If we knew more about the factors determining stem form, we might be in a better position to control it: a cylindrical stem would be an obvious advantage in utilisation! Certainly species, genotype, age, competition, site (especially wind exposure), silvicultural treatment and size and structure of the crown are all important. Of these factors, the crown, particularly crown length, plays a decisive role in determining stem form Cultural measures such as thinning and pruning can markedly alter the incremental growth pattern of individual trees by artificially creating new growth conditions. Thinning converts a stand-grown tree to a simulated open-grown condition, and growth will shift downward on the stem in response to the increased crown size and exposure. Pruning, on the other hand, converts an open-grown or large-crowned tree to a simulated stand-grown condition, and growth will shift upward on the stem in accordance with the new crown position. Again, these responses are not invariable and a change in form often fails to occur following a silvicultural operation. The lack of response in many instances, however, can be traced to the prior condition of the trees, the experimental techniques, or more frequently, to the analysis and interpretation of the results. Inheritance must also be taken into consideration, and it must constantly be born in mind that all trees are predisposed to assume a certain form. Environmental influences and cultural practices can only modify the basic tree form predestined by heredity. Points of inflection along the stem present a major problem in specifying stem form. Besides the two major points, one in the butt region and the other somewhere in or close to the crown base, there are many minor points of inflection along the bole. In other words, tree form is complex. This makes it difficult to specify the profile of a single stem, let alone to specify a profile to represent all stems. Nevertheless, researchers have attempted to do this through two broad approaches: Theoretical - involving investigation of growth processes (distribution of nutrients, cambial activity, water transport, hormone gradients) and deducing what form a tree might take as a result. Empirical - determining stem profiles from empirical evidence and rationalising these as a result of growth processes.
- Tree form is complex. Some general [geometric shapes] approximate portions of the tree bole, but there are many inflections and points of irregularity. The stem does not follow any geometric law hence the form is studied by comparison with a known geometric form. Species and genotype predispose the bole to certain forms, but a wide range of environmental and contextual factors will influence this form.
- In farm forestry the shape of the tree is crucial in determining log value, therefore it is useful to document a summary of the tree's form. One of many different methods for assessing tree form is outlined below:
- In farm forestry the shape of the tree is crucial in determining log value, therefore it is useful to document a summary of the tree's form. One of many different methods for assessing tree form is outlined below:
- In farm forestry the shape of the tree is crucial in determining log value, therefore it is useful to document a summary of the tree's form. One of many different methods for assessing tree form is outlined below: One of the most important form factors in the production of sawlogs is straightness of the butt log. If the tree deviates outside a central axis then the form is likely to be unacceptable for milling purposes or it will severely downgrade log value.
- Metzger proposed that a tree bole should be similar to a cubic paraboloid. He argued that the stem was a beam of uniform resistance to bending anchored at the base, and functioning as a lever arm. A horizontal force on such a beam would exert a strain that increased toward the point of anchorage, and the most economical shape for this beam would be a uniform taper similar to a truncated cubic paraboloid. Gray contended that the tree stem is not firmly anchored to the ground. A quadratic paraboloid shape would be more consistent with the mechanical needs imposed by this assumption. The Mechanistic Theory has received the greatest acceptance to date. It is suited to mathematical analysis and has found applications in practical forestry. Schwendener introduced the theory in 1874, but it was Metzger in a series of classical papers who developed Schwendener's tenets into true mechanistic laws and applied them to silvicultural practices. Metzger proposed that the stem was a beam of uniform resistance to bending, anchored at the base and functioning as a lever arm. A horizontal force will exert a strain on the beam that increases toward the point of anchorage, and if the beam is composed of homogeneous material the most economical shape would be a beam of uniform taper. Metzger demonstrated that this taper approximates the dimensions of a truncated cubic paraboloid (height against diameter^3 is linear). He confirmed his theory for many stems, particularly conifers.
- Various theories, called stem form theories, have been proposed to explain the variations in the way trees accumulate woody material with time. The four main theories are: Nutritional theory Water conducting theory Mechanistic theory or Metzger's beam theory Hormonal theory Support for both the Nutritional and Water Conducting theories has waned in recent decades.
- The above mechanical theories of stem form are one approach towards an explanation of tree shape. Two other theories relate tree bole shape to the need of the tree to tranport water or nutrients within the tree (water conducting theory and nutritional theory of stem form respectively). These theories are based on ideas that deal with the movement of liquids through pipes.The hormonal theory of stem form envisages that growth substances, originating in the crown, are distributed around and down the bole to control the activity of the cambium. These substances would reduce or enhance radial growth at specific locations on the bole and thus affect bole shape. The Hormonal theory envisages growth substances originating in the crown regulating the distribution of radial growth on the stem by controlling the activity of the cambium. It provides a physiological explanation of how a tree grows and why trees differ in the way they do, but it does not specify the particular shapes trees may have under varying circumstances. The hormonal theory offers the most promising approach to the stem form problem. It provides a physiological basis for the nutritional as well as the functional theories of stem form.
- Many attempts to prove and disprove the theory, many experiments supported the theory quite well However, no adequate physiological explanation of the observed facts, theory provides only a realistic interpretation of tree form w.r.t strength requirements & support function
- The Mechanistic Theory has received the greatest acceptance to date. It is suited to mathematical analysis and has found applications in practical forestry. Schwendener introduced the theory in 1874, but it was Metzger in a series of classical papers who developed Schwendener's tenets into true mechanistic laws and applied them to silvicultural practices. Metzger proposed that the stem was a beam of uniform resistance to bending, anchored at the base and functioning as a lever arm. A horizontal force will exert a strain on the beam that increases toward the point of anchorage, and if the beam is composed of homogeneous material the most economical shape would be a beam of uniform taper. Metzger demonstrated that this taper approximates the dimensions of a truncated cubic paraboloid (height against diameter^3 is linear). He confirmed his theory for many stems, particularly conifers.
- Gray (1956), an Australian forester, questioned Metzger's assumption that the tree stem is anchored firmly to the ground, and contended that stem form conforms more closely to the dimensions of a quadratic paraboloid (height against diameter^2 is linear). A stem of this shape would be consistent with the mechanical requirements of a tree, not only to horizontal wind pressure but to other forces acting on the stem. There have been many attempts to prove and disprove the theory. Generally, it is accepted that the theory provides a realistic interpretation of tree form with respect to strength requirements and support function. Though the theory has survived experimental study reasonably well, it does not provide an adequate physiological explanation of the observed facts.
- Metzger proposed that a tree bole should be similar to a cubic paraboloid. He argued that the stem was a beam of uniform resistance to bending anchored at the base, and functioning as a lever arm. A horizontal force on such a beam would exert a strain that increased toward the point of anchorage, and the most economical shape for this beam would be a uniform taper similar to a truncated cubic paraboloid. Gray contended that the tree stem is not firmly anchored to the ground. A quadratic paraboloid shape would be more consistent with the mechanical needs imposed by this assumption. The Mechanistic Theory has received the greatest acceptance to date. It is suited to mathematical analysis and has found applications in practical forestry. Schwendener introduced the theory in 1874, but it was Metzger in a series of classical papers who developed Schwendener's tenets into true mechanistic laws and applied them to silvicultural practices. Metzger proposed that the stem was a beam of uniform resistance to bending, anchored at the base and functioning as a lever arm. A horizontal force will exert a strain on the beam that increases toward the point of anchorage, and if the beam is composed of homogeneous material the most economical shape would be a beam of uniform taper. Metzger demonstrated that this taper approximates the dimensions of a truncated cubic paraboloid (height against diameter^3 is linear). He confirmed his theory for many stems, particularly conifers.
- The stimulus for the determination of form factors was the recognition of the strong resemblance of a tree stem to standard geometric solids. The hope was that ratios would be established which could be used to convert easily computed volumes of standard solids into tree volumes.
- Absolute and normal are no longer used.
- Because form factors are based on volumes (or diameter measurement is needed near the top of the tree as with Pressler), they can not be measured directly or easily on standing trees. For this reason, attempts were made to find a correlation between form factor and some index which could more conveniently be measured on a stem. Such a correlation was found with Form Quotient.
- Form quotient values are grouped into form classes, e.g. Form Class 70 comprises trees with F.Q.s from 0.675 to 0.724
- Lewis et al (1976) use the difference in diameter between 4.5 m or 7.5 m and 1.5 m to estimate the general shape of Pinus radiata in South Australian plantations. This value is used to improve the estimate of stem volume.
- Emphasis on the need for taper equations to produce estimates that were compatible with predictions of stem volume
- Many mensurationists have sought to discover a single, simple two-variable function involving only a few parameters which could be used to specify the entire tree profile. Unfortunately, trees seem capable of assuming an infinite variety of shapes, and polynomials (or quotients of polynomials) with degree at least two greater than the observed number of inflections are needed to specify variously inflected forms. Furthermore, coefficients would vary from tree to tree in ways that could only be known after each tree has been completely measured. Thus, explicit analytic definition (of the tree profile) requires considerable computational effort, yet lacks generality...... Each tree must be regarded as an individual that must be completely measured, or else as a member of a definite population whose average form (profile) can only be estimated by complete measurement of other members of the population selected according to a valid sampling plan.... Hence, polynomial analysis may rationalise observed variation (in the stem profile) after measurement, but it does not promise more efficient estimation procedures."