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F(x) = number of branches

  1. Sep 7, 2015 #1
    1. The problem statement, all variables and given/known data
    A maple tree has branch buds. These grow in groups of 3 at the ends of the branches and in groups of 2 at each year's growth of those branches. The largest branch bud in the groups of 3 just extends the existing branch. Describe this with a function and is the function exponential?

    2. Relevant equations
    2*y years = number of branches from an existing branch

    3. The attempt at a solution
    Year 1: 0
    Year 2: 2
    Year 3: 8
    Year 4: 22

    It looks like it is exponential. But is it really or is it a polynomial function?
     
  2. jcsd
  3. Sep 7, 2015 #2

    andrewkirk

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    Can you describe the process more clearly? It's not possible to understand from what you've written.
    When do the buds appear? When do they start growing into branches? What does 'at each year's growth of those branches' mean?

    Also, your equation 2*y has used the asterisk, which indicates multiplication, yet you say lower down that it looks like an exponential. Did you mean to write an exponential, which is symbolised by ^?

    There is no apparent connection between the formula in 2 and the number series in 3.
     
  4. Sep 7, 2015 #3
    The buds are there all the time. The older ones that extended the branches disappear once the branch has been extended and new buds form.

    They start growing into branches in the spring. "at each year's growth of those branches" means that every year there are 2 branch buds wherever the growth has stopped or is happening that will grow new branches. Sometimes the buds just stay there but often they will grow new branches.

    And no for the branches from existing branch it is supposed to be multiplication. The exponential is for the whole function.

    The connection might be small but there is a connection there. Namely that as the number of branches increase, so do the number of branch buds on existing branches and that it is always by a multiple of 2.
     
  5. Sep 8, 2015 #4

    haruspex

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    The only way I can make sense of this description is to suppose that this statement
    is redundant, having already been implied by the preceding clause in the same sentence.
    On that basis, I can regenerate the sequence you show, except for the 22 at the end. I make it 26.
    If you agree with that, you might find it more instructive to count all the branches in each number, i.e. including the branch you started with.
     
  6. Sep 8, 2015 #5

    mfb

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    Can you draw a sketch of the tree after 0, 1, 2, 3 years?
     
  7. Sep 9, 2015 #6

    RUber

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    Let y_n = branches in year n.
    Then we can say: y_0 = 0, y_1 = 0, y_2 = 2, y_3 = 8, y_4 = 22.
    Assume every existing branch splits into two the following year, then that would look something like:
    y_{n+1} = 2y_n
    Clearly this won't work, since there would be nothing starting the process, it would just be 0, 0, 0, ...
    So there must also be a function of time.
    Something like
    y_{n+1} = 2n
    But that pattern would be too easy to see, it would look like 0, 0, 2, 4, 6, ...
    Maybe your tree problem is a function of both time and existing branches.
     
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