TREE FORM & TREE FORM THEORIES

•Tree form
•Tree form theories
•Form ratios
•Taper equation
 Tree Bole

• Tree bole takes different shapes & tapers generally from base to

tip  dia with ht

Shape ? Taper ?

• Shapes & 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 shape & the tree

crown, bole gets affected by the change in crown

• Shape & branching habit vary with species
 Tree Form

 Tree form is the shape of bole

 Required to know important aspects of form that may affect

marketability
Stem or bole form

• 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
• Complex, not easy to approximate in known geometric
shape in total
Importance of Tree Form

• Knowledge of the tree form can help in

 better estimation of bole volume or biomass

 better understanding of the growth conditions
Perfect tree form

- straight bole
- fine branches
- no apparent defects etc.
Acceptable tree form

- not ideal
- some kinks in stem
- evidence of insect attack etc.
 Unacceptable tree form

- crooked bole
- forked
- evidence of diseases e.g.
rot
 Parts of a tree stem tend to approximate truncated parts

of common shapes :

1. base tends to be neiloid

2. tip tends to be conoid

3. main part of the bole

tends to be paraboloid

 The points of inflection between these shapes, however,

are not constant.

Tree form
The general formula for solid of revolution;               
where, x = distance from the apex of the shape &
y = dia .
Specific values of b correspond to common shapes

1.         Paraboloid where b = 1  y2 = k x  quadratic

where b=0.66  y2 = k x0.66 or y3 = k` x
 cubic
2.       Conoid y = k * x or y = k` * x
2 2

3.      Neiloid y2 = k * x3

What about cylinder?

•Tree form
•Tree form theories
•Form ratios
•Taper equation
Tree form - theories

1. Nutritional theory
2. Water conducting theory
3. Hormonal theory
4. Mechanistic theory or Metzger's beam theory
 Nutritional theory and Water conducting
theory
— based on ideas that deal with the movement of
liquids through pipes
— relate tree bole shape to the need of the tree to
transport water or nutrients within the tree
 The hormonal theory
— growth substances, originates in the crown,
— distributed around and down the bole to control the
activity of the cambium
— these substances reduce or enhance radial growth at
specific locations on the bole and thus affect bole
shape.
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 at which point ?
 On the base so the tree reinforces towards
the base and more material deposited at lower ends
Metzger’s Theory contd…

• A horizontal force will exert a strain on the beam

• Strain increases toward the point of anchorage,
• If the beam is composed of homogeneous material the
most economical shape would be a beam of uniform taper

Tree in open – short, with rapidly tapering boles

Tree in close canopy – long & nearly cylindrical boles
Metzger’s Theory contd…

P
L P = force applied at the free end
L = distance of a given cross
d 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/ d3
 Metzger’s Theory contd…

If W = wind pressure per unit area

A = crown area
Then total pressure P=WxA
S = 32PL/ d3
 S = (32/ ) W A L /d3
For a given tree  d3 = (32/ ) W AL/S  k L
 d3  k L (Cubic paraboloid)

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 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
Metzger’s Theory contd…

• 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…

• 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…

• 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…

Trees growing in dense forest

are subjected to lesser wind

pressure

 longer & cylindrical bole

Tree form
•Tree form
•Tree form theories
•Form ratios
•Taper / Form equation
Form of Tree
• Its Shape
• Shape may be regular but mostly irregular
• Form is measured by different methods
Methods of studying tree form

1. Standard Form ratios

2. Classification of form on
the basis of form ratios

3. Compilation of taper tables

1. Standard Form ratios

 Two Form Ratios are in practice

I. Form factor

II. Form quotient

I. 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

 Classification of Form Factor

Depending upon the height of measurement of basal area

 Absolute form factor – vol. (whole tree) / vol. (cylinder of dia
at ground level)
 Artificial form factor – vol. (whole tree) / vol. (cylinder with

basal area at bh)

 Normal form factor – vol. (whole tree) / vol. (cylinder with

measurement at some % of height)

Absolute form factor
 Defined as the ratio of the volume of the whole tree and the volume of a
cylinder base at above ground level
 For different solid of revolution
neiloid
0.25
cone
0.33
quadratic paraboloid
0.50
cubic paraboloid
0.60
cylinder
1
 Not used
 Artificial form factor or Breast height form factor
– the most common
• 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

Limitations?
Normal 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
 Absolute FF BHFF

Cylinder 1.00 >1

Neiloid 0.25 > 0.25

Conoid 0.33 > 0.33

Quadratic paraboloid 0.50 > 0.50

Cubic paraboloid 0.60 > 0.60

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.

II. Form quotient

• Ratio of the diameter at two different

places on the tree

• Generally calculated for,
d
diameter, d(some point above bh) to DBH
• Absolute form quotient – most common

DBH
 Form quotient contd…

Absolute form quotient

• Calculated by measuring the dia at

a height halfway between bh and

total tree height
• Absolute form quotient = dia at

halfway/ dbh
• Commonly expressed as a decimal

e.g. 0.70
• Grouped into form classes
2. Classification of form on the basis of
form ratios

i. Form class
 Defined as one of the intervals in which the range of
Form Quotients of trees is divided for classification and
use
 Trees may be grouped into form classes expressed as
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 …

Form class & quotient contd…

Absolute form quotients also suggest general stem shapes:

Neiloid 0.325 - 0.375 (FQ class 35)

Conoid0.475 - 0.525 (FQ class 50)

Quadratic paraboloid 0.675 - 0.725 (FQ class 70)

Cubic paraboloid 0.775 - 0.825 (FQ class 80)

Form Point
- Focal point of wind force.
- Located approximately at the centre of gravity of
crown since crown offers max. resistance

ii. 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)
3. Compilation of Taper tables

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 Table

 Taper Table portrays stem form in such a way that the data
can be used in calculation of stem volume
 if sufficient diameters are taken at successive pts. along
stem, taper tables can be prepared
Kinds of Taper Table :

1. Diameter Taper Table : gives taper directly for

dbh without referring to the tree form

2. Form Class Taper Table : Dia at different fixed

points on the stem expressed as % of dbh (ub) for different
form classes
•Tree form
•Tree form theories
•Form ratios
•Taper equation
 Taper equations
 Taper equations attempt to describe taper as a function of tree

variables such as dbh, height, etc.

 Try to fit the stem profile in model form and predict the dia at

any point on the tree stem

 Numerous approaches for the construction of these equations

such as:
 Use of computers to fit complex polynomial equations to describes the
continuous change in stem form from ground to tip
 Often the tree bole was segmented into 2 or 3 parts and separate
equations fitted to each part
 Taper equations contd…

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…

Can be used to provide

 predictions of dia at any point on the stem
 estimates of total stem volume
 estimates of individual log volumes
 Taper equations contd…

1. 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 top of the tree to the point at
which ‘d’ is measured
2. Hojer’s Formula

d/dbh = C log [(c+L)/c]

d is Dia at ‘L’ (same as previous)

C & c are constants.

“Trees assume infinite variety of shape”-- explicit analytic

definition of tree form requires considerable computational
effort -- yet lacks generality”.
 Form Factors
 Form Quotient
 Form Class
 Taper table
 Taper equations
 Volume of Trees
From of Solid Volume of Volume of a frustum Remarks
Full solid of solid

1 Cylinder Sl Sl -

2 Paraboloid S l i) S 1 + S2 X l Smalian’s
2 2 formula

ii) SmX l Huber’s formula

3 Cone S l S 1 + S2+ √(S1 . S2 )

3 X l

3
4 Neiloid Sl ( S 1 + 4Sm+ S2 ) X l Prismoidal or
4 6 Newton’s
formula
Q.

Calculate the volume of a tree 13 m high having the

following measurements: Diameters at B.H., 4.24m, 7.24m
and 8.74 m are 40 cms, 32 cm, 20 cm and 15 cm respectively.
8.74
7.24
5.74
4.24
2.74
1.37
S.No. Dia Dia in Basal Area Length of Volume(m3)
measured cm (m2) section (m)
at ht. (m)

(d) B=(d2)/4 H

(1) 1.37 40 2.74 (B*H)

(2) 4.24 32 3.00 (B*H)

(3) 7.24 20 3.00 (B*H)

(4) 8.74 (cone) 15 4.26 (1/3B*H)