Average Age of Trees in Town: Solved!

[meteorologist1]

Hi, I was curious how one would solve this problem:

In a town, there are 1850 trees along public roads. Each year, the town has to remove on average 25 trees of random age because of various reasons (natural death, fungus infection, insects, hit by cars, roots damaged by construction, etc.). To compensate for the loss, the town plants on average 25 new trees (age = 0) to keep the tree stock the same. Over time, what is the average age of the tree stock?

 

[NateTG]

I think the best way to go about it is to identify the steady state solution and then check to make sure that it's stable. Aging increases the average age by a predictable amount every year. The cutbacks need to keep up with that.

 

[meteorologist1]

Yes. I think the tricky part to this problem is mathematically describing the removal of the 25 trees -- there's some of probability involved here. A tree that is removed could be one that was planted the year before or it could be 100 years old.

 

[uart]

Yes. I think the tricky part to this problem is mathematically describing the removal of the 25 trees -- there's some of probability involved here. A tree that is removed could be one that was planted the year before or it could be 100 years old.

Yes you have make some assumptions about the probabilty distribution of the removed trees otherwise you can't do the problem. I think they just expect you to assume that the age of the removed trees is uniformly distributed. Actually any distribution that is symmetrical about the mean will yeild the same result, but uniform is the simplist assumption.

Do you know how to write a difference equation for the average age (that is, something like x(k+1) = a x(k) + b.

BTW the answer I got is 73 years. See if you can get the same answer with the above hints.

 

[meteorologist1]

I was thinking the following: Because you plant 25 new trees each year, over time, the average age of the tree stock would be equivalent of that having 25 trees of each age ranging from age 0 to age (1850/25) - 1 = 73. Because sometimes you remove a young tree and sometimes a very old tree, over time, things will average out. Therefore, I actually obtain an average age of 36.5 years old using that assumption.

 

[climber/jumper]

I was thinking the following: Because you plant 25 new trees each year, over time, the average age of the tree stock would be equivalent of that having 25 trees of each age ranging from age 0 to age (1850/25) - 1 = 73. Because sometimes you remove a young tree and sometimes a very old tree, over time, things will average out. Therefore, I actually obtain an average age of 36.5 years old using that assumption.

What's to say a tree can't be older than 73?

 

[HallsofIvy]

What's to say a tree can't be older than 73?

Meteorologist1 is arguing that, since each year we add 25 trees of age 0, while "culling" 25 trees, in the long run we will have 25 trees of age 0, 25 trees of age 1, etc. up to the limit of 1850 trees. If the 25 trees culled were always the oldest 25 trees, that might be valid but I'm not sure it is in general.

 

[meteorologist1]

Yes, the key is in the long run. I'm saying that the "net effect" would be like having that situation I described. You can have a tree older than 73. I'm trying to argue that if you cut 25 trees a year ranging from very young to very old (older than 73), their average age would be around the age of 73 -- so it's like having cut down 25 trees of age 73. That's my guess. uart, you can explain your answer?

 

[climber/jumper]

Yes, the key is in the long run. I'm saying that the "net effect" would be like having that situation I described. You can have a tree older than 73. I'm trying to argue that if you cut 25 trees a year ranging from very young to very old (older than 73), their average age would be around the age of 73 -- so it's like having cut down 25 trees of age 73. That's my guess. uart, you can explain your answer?

Hint: probability that a tree gets cut down in their first year is:
The number of ways to choose 25 trees such that this tree is chosen/ the number of ways to choose 25 trees. (Both with replacement)

call this probability "p"

the probability that a tree will not get cut down in any given year will be 1-p
the probability that a tree survives the first 2 years will be (1-p)^2.
what is the probability the tree gets cut down on the second year? 1-(1-p)^2

Let X be the number of years a tree survives, "what is the average length of life of a tree?" is equivalent to "What is the expected value of X?"

Hope this helps.

 

[Office_Shredder]

I was thinking the following: Because you plant 25 new trees each year, over time, the average age of the tree stock would be equivalent of that having 25 trees of each age ranging from age 0 to age (1850/25) - 1 = 73. Because sometimes you remove a young tree and sometimes a very old tree, over time, things will average out. Therefore, I actually obtain an average age of 36.5 years old using that assumption.

This actually requires them to cull the 25 oldest trees each year (imagine you start with 1850 trees at age 0, and cull 25 of the oldest each year. You'll reach this state eventually). You admit that there can be a set of trees with a higher average age, and yet, there can't be a set of trees with a younger average age. So, this is a minimum, not an average

 

[NateTG]

Yes, the key is in the long run. I'm saying that the "net effect" would be like having that situation I described. You can have a tree older than 73. I'm trying to argue that if you cut 25 trees a year ranging from very young to very old (older than 73), their average age would be around the age of 73 -- so it's like having cut down 25 trees of age 73. That's my guess. uart, you can explain your answer?

Consider the total of the ages of the trees. Every year, that number increases by 1850, since there are 1850 trees. Therefore, the averate total age of the trees that are removed must also be 1850 for the population to be in a steady state. Thus the average age of the trees that are removed must be 73 years.

Now, if we assume that the 25 trees are chosen from the general tree population at random with an even distribution, then the average age of the trees will be 73 years. For biased random selections of trees, the average age of the population can range from 36.5 years to unbounded.

 

[uart]

That's my guess. uart, you can explain your answer?

Let x(k) be the total age of all the 1850 trees at year k, so the average age of the trees at any year is x(k)/1850.

For any symmetrical probabilty distribution the average age of the removed tress is equal to the average age of the enitire population.

So x(k+1) = x(k) - 25 * x(k)/1850 + 1825,
which gives, x(k+1) = 1825/1850 * x(k) +1825.

This is the difference equation. Note that the subtracted term, 25x(k)/1850, is due to the 25 removed tress and the additive term (1825) is due one year aging of the non-removed trees.

In steady state x(k+1) = x(k), so the equation becomes x(k) (1 - 1825/1850) = 1825 which gives the steady state solution of x = 1825/25 * 1850. That's the total age of all the tress remember, so you have to divide it by 1850 to get the average age of 73.

 

[uart]

Actually I just noticed in the above solution that I probably should have added one year for all the 1850 trees, not just for the 1825 non-replaced trees. This would especially be true if all dead trees were replaced immediately when they died, or alternatively if the age of the dead trees counted equally to the entire age of the plantation - both of which are reasonable assumptions. So replacing the 1825 with 1850 in the additive term of the above difference equation changes the solution from 73 years to 74 years.