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Another useful measurement to indicate tree vigor is live crown ratio (LCR). It is the ratio of crown length to total tree height, or the percentage of a tree’s total height that has foliage (Figure 5.6).
Figure 5.6. Live Crown Ratio: the ratio of live crown to total tree height; expressed as a percentage.
Crown length is partly a function of species’ shade tolerance. For example, Douglasfir and most pines will selfprune (drop their lower branches as they become shaded). However, a shade tolerant species such as western hemlock will keep its lower branches in medium shade. Therefore, a western hemlock will have a longer crown (and higher LCR) under low light conditions than a Douglasfir. Consider the data in Table 5.2. This young, evenaged forest stand (≈ 40 yrs), had a substantial riparian component supporting the hardwoods, and was growing on a southern slope. Note that the red alder, cherry and Douglasfir had shorter LCR’s than the more shade tolerant hemlock and cedar. The hardwoods were shorter, but all the conifers were approximately the same height.
Table 5.2. Mean live crown ratio (LCR) and height by species for an evenaged stand in the Latourell Creek watershed. Data collected by Mt. Hood Community College Forest Measurements I class March, 2003.  
Species  LCR(%)  HT (ft.) 
Red alder  40  62 
Bitter cherry  28  52 
Douglasfir  43  74 
Western redcedar  74  80 
Western hemlock  64  78 
Douglasfir trees with large crown ratios (>50%) tend to be dominant trees, and/or trees growing with adequate light. Douglasfir trees with ratios less than 30% generally have low vigor, and typically either a) occupy intermediate or suppressed crown classes, or b) are growing in very dense, uniform young stands. In the latter case, their root systems do not develop well, and the trees become subject to windthrow over time. These “dog hair stands” are often a result of planting seedlings at a high density, and failing to thin them later at the appropriate time.
In general, LCR will reflect crown class, regardless of species. Trees growing in the dominant crown classes tend to have the longest crowns overall, followed by trees in the codominant, intermediate, and suppressed crown classes respectively (Table 5.3). The exception to this may be unevenaged or twoaged stands in which distinct second and third layers are composed primarily of shade tolerant trees. In these cases, each layer must be evaluated independently.
Table 5.3. Mean live crown ratio (LCR) for species in an evenaged BLM stand near Larch Mountain. Data collected by MHCC Forest Measurements I class January 2003.  
Mean LCR by Crown Class  
Species  D (%)  C (%)  I (%)  S (%) 
Douglasfir  48  47  38  
Western hemlock  50  37  39  35 
Overall  49  42  38  35 
5.3: Live Crown Ratio  Biology
Because each whorl represents one year of growth, age on young trees with determinate height growth can be estimated by counting the whorls.
1. On most trees, the lowest tree branches are systematically dropped as the tree grows and the sun no longer hits the base of the tree. Therefore, when estimating age using this method, it is important to include the bottommost stubs and/or knots where it is evident branches once existed.
2. Two to four years should be added to most species to allow for the time between seedling germination and evidence of branch whorls on the trunk (Figure 4.4).
3. Small single branches between major branch whorls do not constitute a true whorl or year of growth. Do not count these false whorls.
4. A very short increase in length between whorls that seems unlike the other years’ growth may indicate a “lammas” year, in which the tree flushed twice, often in response to extraordinary growing conditions. Ignore those years unless it is evident that some injury is responsible for the very short internode (Figure 4.4).
Figure 4.4. Counting the whorls to determine age of a young conifer. Lammas growth and false whorls are ignored. The lower stem is examined for knots, and years to first visible knot is estimated and added in — generally 24 years.
This method of “counting the whorls” usually works very well up to fifteen years of age or so for conifers such as Douglasfir, spruces (Picea spp.), pines (Pinus spp.) and true firs (Abies spp.). It is more challenging for cedars (Thuja spp., Chamaecyparis spp.), hemlocks (Tsuga spp.), and some hardwoods. One really has to get close to the tree, look carefully for evidence of bud scars, and know the growth habits of these species.
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Effect of Frame Size on Ideal Weight
Your body frame size helps determine your IBW, too. The equation calculates the IBW for someone with a medium frame. You can subtract 10 percent for a person of the same height with a small frame, and add 10 percent for a largeframed individual. So in the end, you end up with an IBW range.
For example, for the 5foot, 4inch tall woman in the first example with an IBW of 120 pounds, her IBW range is 108 to 132 pounds. A healthy weight for smallframed woman of this height would be toward the low end of 108 pounds, while a largeframed woman could weigh toward the high end. And the man who's 6 feet tall has an IBW range of 160 to 196 pounds.
If you're not sure of your frame size, here's a quick way to figure it out. Place your thumb and middle finger around your wrist, right where you'd wear a watch. If your fingers overlap, you have a small frame. If they touch, your frame is medium and if they don't meet, you have a large frame.
Caring for a New Lawn
To prevent drying of planting material, keep the top 1.5 inches of the soil moist. This may require light watering two or three times a day for 7 to 21 days. Bluegrass takes 7 to 14 days longer to germinate than other coolseason grasses. As the seedlings grow and root, water less often but for longer periods. For mixtures containing bluegrass, do not make the mistake of decreasing water as soon as the seedlings appear. Continue watering until the bluegrass seedlings emerge. After the third mowing, water to a depth of 6 to 8 inches about once a week or when needed.
Begin mowing as soon as the grass is 50 percent higher than the desired height. For example, mow tall fescue back to 3 inches when it reaches 4.5 inches. The frequency of mowing is governed by the amount of growth, which depends on temperature, fertility, moisture conditions, the season, and the natural growth rate of the grass. The suggested height of cut is given in Table 1. The homeowner should cut often enough that less than onethird of the total leaf surface is removed. Use a mower with a sharp blade. To reduce the danger of spreading disease and injuring the turf, mow when the soil and plants are dry. If clippings are heavy enough to hold the grass down or shade it, catch them or rake and remove them. Otherwise, do not bag the clippings. Allow them to fall into the turf where they will decay and release nutrients. This may reduce the need for fertilizer by 20 to 30 percent.
Pest Control
Fungicides and insecticides are rarely needed on new lawns, and different planting methods require different pest control methods. If pesticides are used, always read and follow label directions.
Seeding. Siduron (Tupersan 50WP) may be applied to coolseason grasses at the time of spring seeding for selective preemergence control of some annual grassy weeds like crabgrass. Other herbicides may be applied to young seedlings during establishment. Get the latest recommendations by visiting the NC State TurfFiles website.
Broadleaf weeds are common in new seedings. However, many will be controlled with frequent mowing at the proper height. After the lawn has been mowed three times, remaining weeds may be controlled using the minimum label rate of a broadleaf herbicide. The particular herbicide used depends upon the weeds present and the tolerance of the turfgrass to the herbicide.
Spaceplanting sprigs, broadcasting sprigs, and plugging. Atrazine (AAtrex) or simazine (Princep) may be applied for control of certain annual grass and broadleaf weeds when sprigging bermudagrass, centipedegrass, St. Augustinegrass, and zoysiagrass. Do not apply these herbicides over the rooting areas of trees and ornamentals that are not listed as being tolerant on the herbicide label.
Sodding. Preemergence herbicides, such as siduron (Tupersan) and bensulide (Betasan), can be applied for annual weedy grass control after sodding cool and warmseason grasses.
Always Fibonacci?
I remember as a child looking in a field of clover for the elusive 4leaved clover  and finding one.A fuchsia has 4 sepals and 4 petals:  and sometimes sweet peppers don't have 3 but 4 chambers inside: 
and here are some flowers with 6 petals:
crocus  narcissus  amaryllis 
Here are some more examples of nonFibonacci numbers:
Here is a succulent with a clear arrangement of 4 spirals in one direction and 7 in the other:  and here is another with 11 and 18 spirals:  whereas this Echinocactus Grusonii Inermis has 29 ribs: 

So it is clear that not all plants show the Fibonacci numbers!
Another common series of numbers in plants are the Lucas Numbers that start off with 2 and 1 and then, just like the Fibonacci numbers, have the rule that the next is the sum of the two previous ones to give:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..
But, no matter what two numbers we begin with, the ratio of two successive numbers in all of these Fibonaccitype sequences always approaches a special value, the golden mean, of 1.6180339. and this seems to be the secret behind the series. There is more on this and how mathematics has verified that packings based on this number are the most efficient on the next page at this site.
 A sunflower with 47 and 76 spirals is an illustration from:
 Quantitative Analysis of Sunflower Seed Packing by G W Ryan, J L Rouse and L A Bursill, J. Theor. Biol. 147 (1991) pages 303328
 Variation In The Number Of Ray And DiscFlorets In Four Species Of Compositae P P Majumder and A Chakravarti, Fibonacci Quarterly 14 (1976) pages 97100.
In this article two students at the Indian Statistical Institute in Calcutta find that "there is a good deal of variation in the numbers of rayflorets and discflorets" but the modes (most commonly occurring values) are indeed Fibonacci numbers.
A quote from Coxeter on Phyllotaxis
it should be frankly admitted that in some plants the numbers do not belong to the sequence of f's [Fibonacci numbers] but to the sequence of g's [Lucas numbers] or even to the still more anomalous sequences
Thus we must face the fact that phyllotaxis is really not a universal law but only a fascinatingly prevalent tendency .
 He cites A H Church's The relation of phyllotaxis to mechanical laws, Williams and Norgate, London, 1904, plates XXV and IX as examples of the Lucas numbers and plates V, VII, XIII and VI as examples of the Fibonacci numbers on sunflowers.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
References and Links
 Mathematical Mystery Tour by Mark Wahl, 1989, is full of many mathematical investigations, illustrations, diagrams, tricks, facts, notes as well as guides for teachers using the material. It is a great resource for your own investigations.
 Fascinating Fibonaccis by Trudi Hammel Garland.
This is a really excellent book  suitable for all, and especially good for teachers seeking more material to use in class.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
5.3: Live Crown Ratio  Biology
1. Exponential Population Growth: N = N_{o}e rt
Table 1A. Final population size with given annual growth rate and time.Be sure to enter the growth rate as a decimal (for example 6% = .06).
[ JavaScript Courtesy Of Shay E. Phillips © 2001 Send Message To Mr. Phillips ]
T he above Table 1 will calculate the population size (N) after a certain length of time (t). All you need to do is plug in the initial population number (N _{o} ), the growth rate (r) and the length of time (t). The constant (e) is already entered into the equation. It stands for the base of the natural logarithms (approximately 2.71828). Growth rate (r) and time (t) must be expressed in the same unit of time, such as years, days, hours or minutes. For humans, population growth rate is based on one year. If a population of people grew from 1000 to 1040 in one year, then the percent increase or annual growth rate is 40/1000 x 100 = 4 percent. Another way to show this natural growth rate is to subtract the death rate from the birth rate during one year and convert this into a percentage. If the birth rate during one year is 52 per 1000 and the death rate is 12 per 1000, then the annual growth of this population is 52  12 = 40 per 1000. The natural growth rate for this population is 40/1000 x 100 = 4%. It is called natural growth rate because it is based on birth rate and death rate only, not on immigration or emigration. The growth rate for bacterial colonies is expressed in minutes, because bacteria can divide asexually and double their total number every 20 minutes. In the case of wolffia (the world's smallest flowering plant and Mr. Wolffia's favorite organism), population growth is expressed in days or hours.
T here are more than 230,000 species of described flowering plants in the world, and they range in size from diminutive alpine daisies only a few inches tall to massive eucalyptus trees in Australia over 300 feet (100 m) tall. But the undisputed world's smallest flowering plants belong to the genus Wolffia, minute rootless plants that float at the surface of quiet streams and ponds. Two of the smallest species are the Asian W. globosa and the Australian W. angusta . An average individual plant is 0.6 mm long (1/42 of an inch) and 0.3 mm wide (1/85th of an inch). It weighs about 150 micrograms (1/190,000 of an ounce), or the approximate weight of 23 grains of table salt. One plant is 165,000 times shorter than the tallest Australian eucalyptus ( Eucalyptus regnans ) and seven trillion times lighter than the most massive giant sequoia ( Sequoiadendron giganteum ). T he growth rate for Wolffia microscopica may be calculated from its doubling time of 30 hours = 1.25 days. In the above population growth equation (N = N _{o} e rt ), when rt = .695 the original starting population (N _{o} ) will double. Therefore a simple equation (rt = .695) can be used to solve for r and t. The growth rate (r) can be determined by simply dividing .695 by t (r = .695 /t). Since the doubling time (t) for Wolffia microscopica is 1.25 days, the growth rate (r) is .695/1.25 x 100 = 56 percent. Try plugging in the following numbers into the above table: N _{o} = 1, r = 56 and t = 16. Note: When using a calculator, the value for r should always be expressed as a decimal rather than a percent. The total number of wolffia plants after 16 days is 7,785. This exponential growth is shown in the following graph where population size (Yaxis) is compared with time in days (Xaxis). Exponential growth produces a characteristic Jshaped curve because the population keeps on doubling until it gradually curves upward into a very steep incline. If the graph were plotted logarithmically rather than exponentially, it would assume a straight line extending upward from left to right. W olffia plants have the fastest population growth rate of any member of the kingdom Plantae. Under optimal conditions, a single plant of the Indian species Wolffia microscopica may reproduce vegetatively by budding every 30 hours. One minute plant could mathematically give rise to one nonillion plants or 1 x 10 30 (one followed by 30 zeros) in about four months, with a spherical volume roughly equivalent to the size of the planet earth! Note: This is purely a mathematical projection and in reality could never happen!
The following illustration shows a comparison of the size of one minute wolffia plant, roughly intermediate between a water molecule and the planet earth!
2. Population Growth Shown As A Geometric Progression A geometric progression is a simplified way to show exponential population growth. Starting with one couple, assume that every female has 4 children (2 boys and 2 girls). The following table compares the population growth in 7 generations. The original couple has 4 children, two of which are girls which give rise to 8 children (2 x 4). Four of the 8 children are girls which give rise to 16 children (4 x 4), etc. This is an exponential increase in which the population doubles each generation. The 7th generation has a population of 2 7 or 128. HeighttoFoot Ratio and Growth SpurtsUnless you're a forensic scientist, you probably don't need to know how to use a complicated heighttofoot equation to figure out height. But looking at relative foot size in growing children size can give you some insight into their future growth patterns, since growth spurts tend to start in the extremities before a child sees overall growth. One study, published in Scoliosis in 2001, found that girls experienced fast growth in their feet a little over a year before hitting their growth spurt during puberty, and boys experience foot growth 2 1/2 years before their major growth spurt. The takeaway? If your children are quickly outgrowing their shoes, that's a major sign that they're headed for a growth spurt in the next one to three years. AbstractCoronavirus infections have emerged as epidemic and pandemic threats in last two decades. After the H1N1 influenza pandemic in 2009, recently diagnosed novel betacoronavirus or severe acute respiratory syndrome coronavirus (SARSCoV)2 has spread across 203 countries and territories in all 5 major continents. World Health Organization (WHO) declared this as a public health emergency of international concern on January 30, 2020. Subsequently on February 11, 2020 a new name was given to this disease i.e. COVID19 by an expert group from WHO. As of April 12, 2020, 10:00 CET, GMT+2:00, 1,696,588 confirmed cases and 105,952 confirmed deaths have been reported to the WHO. (Coronavirus disease 2019, situation report 83). It possibly originated from a small animal market in Wuhan, China. A cluster of patients were admitted with unusual pneumonia not responding to treatment in various hospitals. Epidemiological, genomic analysis and correlation with other coronaviruses led to the isolation of new coronavirus, closely resembling the bat coronaviruses, from such patients in Wuhan. They were identified as the SARSCoV2. This virus infection presents as influenza like illness in the affected people. Fever, cough, respiratory distress with fatigue, diarrhea, nausea and vomiting are common symptoms seen in adults. This may progress on to severe respiratory distress, hypoxia, need for oxygen supplementation and ventilator support as seen in patients in the SARSCoV1 epidemic (2003) in Guangdong, China. The transmissibility of SARSCoV1 was less as compared to SARSCoV2 infection, and it was well controlled with good public health efforts. The present COVID19 epidemic is still in the acceleration phase of 3 and 4 in various countries. Without any effective antiviral agents available at present, the need of the hour is early case detection, isolation of cases, use of good preventive care measures by the household contacts and in the hospital set up. The results of ongoing clinical trials on hydroxychloroquine, azithromycin alone or in combination and a new antiviral agent remdesivir may help to treat some of the infections. A need for effective vaccine is being seen an as good preventive strategy in this pandemic. However the results of clinical trials and incorporation of vaccines in public health programs is a long way to go. Marine Fish Larviculture RequirementsFatty Acid NutritionMarine fish larvae require live feeds that contain essential nutrients at appropriate concentrations. One group of essential nutrients are the fatty acids, organic acids found in animal and vegetable fats and oils. Fatty acids are mainly composed of long chains of hydrocarbons (molecules containing carbon and hydrogen) that end with a carboxyl group (comprised of carbon, two oxygen atoms, and hydrogen). Fatty acids are considered saturated when the bonds between carbon atoms are all single bonds and are unsaturated when some of these bonds are double bonds. Fatty acids have double bonds that start at carbon number 0, 3, 6, or 9. The process of increasing the number of carbons in a fatty acid is termed elongation increasing the number of double bonds is termed desaturation. As an example, the fatty acid eicosapentaenoic acid (EPA, 20:5n3) has 20 carbons and 5 double bonds, and the first double bond is on the third carbon atom. Elongation will increase the number of carbons to greater than 20 and desaturation will increase the number of double bonds to more than 5. Most organisms cannot efficiently change the location of the first double bond so n3 fatty acids cannot be converted to n6. (The "n3 fatty acids" are also known as "&omega3" or "omega3" fatty acids, and the "n6 fatty acids" are also known as "&omega6" or "omega6" fatty acids.) The n3 highly unsaturated fatty acids (HUFAs) docosahexaenoic acid (DHA, 22:6n3) and eicosapentaenoic acid (EPA, 20:5n3) are essential for marine fish (Watanabe 1993). The ratio of DHA to EPA significantly affects the survival of marine fish larvae. The yolk of many wild marine fish eggs contain a DHA:EPA ratio of about 2.0, which suggests at least a 2:1 ratio of DHA:EPA in firstfeeding larvae (Parrish et al. 1994). The ability to synthesize EPA, and subsequently DHA, through elongation of linolenic acid (LNA, 18:3n3) is absent in most tropical and subtropical marine fish. Therefore, they must rely on their diet to receive these essential nutrients. Marine fish contain large amounts of DHA and EPA in the phospholipids of their cellular membranes, specifically in the neural and visual membranes (Sargent et al. 1999). A lack of these essential fatty acids can result in retarded physiological development and altered behavior, such as impaired pigmentation and poor vision in low light intensities, resulting in increased vulnerability to predation and reduced hunting capability (Bell et al. 1995 Estevez et al. 1999 Sargent et al. 1999). A similar situation exists for marine fish larvae and the n6 fatty acids. Arachidonic acid (ARA, 20:4n6) cannot be synthesized from linoleic acid (LA, 18:2n6) by many marine fish species. ARA is a precursor to the eicosanoids, an important group of immunological compounds, which includes prostaglandins and leukotrienes. Without these compounds, the fish immune system is compromised. However, overenrichment of ARA could have deleterious effects, so an optimal ratio for the species of interest should be maintained (Bessonart et al. 1999 Estevez et al. 1999). Physical Characteristics of PreyThe size of live feed organisms and their ability to elicit a feeding response from fish larvae are important considerations in marine fish larviculture. The small mouth gape of many marine fish larvae limits the size of food it can consume and prevents the initial use of larger live food organisms such as brine shrimp. As fish larvae have evolved to feed on natural congregations of zooplankton, the stimuli produced by the movement of live feed organisms is needed for many marine fish larvae to elicit a feeding response. Larval mouth gape and feeding response to various live feeds are species specific both should be established for the species to be cultured since they will determine which live food to use. Interval Scale Examples, Definition and MeaningIn the world of data management, statistics or marketing research, there are so many things you can do with interval data and the interval scale. With this in mind, there are a lot of interval data examples that can be given. On this page you will learn:
As you might know, there are 4 measurement scales: nominal, ordinal, interval, and ratio. Knowing the measurement level of your data helps you to interpret and manipulate data in the right way. Let’s define the interval data: Interval data refers not only to classification and ordering the measurements, but it also specifies that the distances between each value on the scale are equal. The distance between values is meaningful. To put it another way, the differences between points on the scale are equivalent. That is why it is called interval data. It is measured on interval scales. The Interval scale is a numeric scale that represents not only the order but also the equal distances between the values of the objects. The most popular example is the temperature in degrees Fahrenheit. The difference between a 100 degrees F and 90 degrees F is the same difference as between 60 degrees F and 70 degrees F. Time is also one of the most popular interval data examples measured on an interval scale where the values are constant, known, and measurable. These characteristics allow interval data to have many applications in the statistics and business intelligence field. However, there is one major disadvantage – the lack of absolute zero. In the interval scale, there is no true zero point or fixed beginning . They do not have a true zero even if one of the values carry the name “zero.” For example, in the temperature, there is no point where the temperature can be zero. Zero degrees F does not mean the complete absence of temperature. Since the interval scale has no true zero point, you cannot calculate Ratios . For example, there is no any sense the ratio of 90 to 30 degrees F to be the same as the ratio of 60 to 20 degrees. A temperature of 20 degrees is not twice as warm as one of 10 degrees. Thus, interval scale only allows you to see the direction and the difference between the values, but you can not make statements about their proportion and correlation. So let’s sum the key characteristics of the interval data and scales:
So, interval scales are great (we can add and subtract to them) but we cannot multiply or divide. In addition, in the practice, many statisticians and marketers can turn a noninterval ordered values scale into an interval scale to support statistical or data analysis. Interval data examples: 1. Time of each day in the meaning of a 12hour clock. 2. Temperature, in degrees Fahrenheit or Celsius (but not Kelvin). 3. IQ test (intelligence scale). 4. Test scores such as the SAT and ACT test scores. 5. Age is also a variable that is measurable on an interval scale, like 1, 2, 3, 4, 5 years and etc. 6. Measuring an income as a range, like $999 $1000$1999 $2000$2900, and etc. This is a classic example of turning a noninterval, ordered variable scale into an interval scale to support statistical analysis. 7. Dates (1015, 1442, 1726, etc.) 8. Voltage e.g. 110 and 120 volts (AC) 220 and 240 volts (AC) and etc. 9. In marketing research, if we ask 2 people how much time do they spend reading a magazine each day, we would know not only who spend more time reading but also the exact difference in minutes (or another time interval) between the two individuals. 10. Grade levels in a school (1st grader, 2nd grader, and etc.) Interval and Ratio Data Understanding the difference between interval and ratio data is one of the key data scientist skills. Interval and ratio data are the highest levels of data measurements. But still, there is important differences between them that define the way you can analyze your data. As the interval scales, Ratio scales show us the order and the exact value between the units. However, in contrast with interval scales, Ratio ones have an absolute zero that allows us to perform a huge range of descriptive statistics and inferential statistics. The ratio scales possess a clear definition of zero. Any types of values that can be measured from absolute zero can be measured with a ratio scale. The most popular examples of ratio variables are height and weight. In addition, one individual can be twice as tall as another individual. When it comes to the possibility of analysis, Ratio scales are the king. The variables can be added, subtracted, multiplied, and divided. So, with ratio data, you can do the same things as with interval data plus calculating ratios and correlations. Examples of ratio data:
In addition, ratio and interval data are both quantitative data. So, both might also be classified as Discrete or Continuous. See our post discrete vs continuous data. In many types of research such as marketing research, social, and business research, interval and ratio scales represent the most powerful levels of measurements. Watch the video: Measuring the canopy of a tree (May 2022). 