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Treebeard's Stumper Answer
2 February 2001

Paper Trees

In last week's Dunn Middle School Friday Newsnote, Chris Beebe found that just one local supermarket uses over two million paper bags per year. Chris wondered how many trees that was. I had the same thought on that Saturday morning last July when the UPS truck delivered my copy of the (then) brand-new Harry Potter and the Goblet of Fire. That was a record printing of 3.8 million books. It's a hefty volume that measures 6 1/4 by 9 1/4 by 2 inches and weighs over 2 1/2 pounds. How many trees is that? How much forest? Estimating an answer will require intelligent assumptions, some guessing, and some research!

Like half a million other readers, I placed my advance order for a copy of the fourth Harry Potter book from Many kids got their copies at special midnight book parties held at book stores across the country. It was an event! The UPS truck (or was it FedEx?) pulled up early on Saturday morning, July 8, 2000. I peeked in the back of the truck at a huge pile of books from Amazon, and nothing else. I wish I had my camera ready. However you feel about the story, it's great to see the kids so excited by any book. Check out the DMS Book Review page for other interesting books our bright kids are reading on their own.
3.8 million books...  

(Maybe there's something fishy about my question?
But estimate anyway, as an upper bound at least!
) how much forest?

I took an average tree to be a foot in diameter and 50 feet high, a tall utility pole. I weighed a 2x4 on the bathroom scale and crunched some numbers to figure volume and weight. I estimate that one tree yields about 200 books, so the 3.8 million Harry Potter books come to 19,000 trees. With ten foot spacing, that's a forest a bit over a quarter mile on a side, less than I expected. The details are below. Of course your answer will vary with your own assumptions. What a great resource a forest is! It can bring both culture and habitat, if we can just maintain a sustainable balance.


Usually when we ask "How many trees did that take?", we ironically suggest it wasn't worth it. I suggest nothing. How many trees did that Harry Potter book printing really take? I made several estimates using different assumptions, and I was impressed how close they actually came out. I do have some second thoughts about this question that I'll consider later on. Here's how I figured my estimate, showing all my work of course.

Estimate #1: figure by weight

I figured an average tree to be a foot in diameter and 50 feet high, like a tall utility pole. Ignoring taper and branches and bark, the volume is about

    volume = pi r2h 
    (tree) = pi x 6 x 6 x 50 x 12
           = 67,858 cubic inches
I stood on a bathroom scale with a dry 8 foot 2x4 (actually 1.3 inch x 3.5 inch), and found it weighs about 8 pounds. (Remember, this is just an estimate!) The volume of that 2x4 is
    volume = length x width x height
    (2x4)  = 1.5 x 3.5 x 8 x 12
           = 504 cubic inches
So the tree is the equivalent of 67,858 / 504 = 135 2x4s. At 8 pounds apiece, that comes to 135 x 8 = 1077 pounds of dry wood.

Logs are debarked and chipped and then "cooked" in a chemical process. Lignin and other unwanted parts of the wood can then be removed, leaving a pure cellulose pulp. states that

In the conventional sulfite cook using softwood, the typical yield is 44 to 46 percent, based on wood and with a lignin content of 2 to 5 percent. At that point, a relatively light-coloured pulp with good strength properties is obtained, suitable for use in the unbleached state, especially in mixture with groundwood for a variety of printing papers. For pulps in which high brightness (whiteness) is desired, the residual lignin is removed by bleaching
There are other chemical methods, but I stuck with that average 45% figure. A 45% yield from 1077 pounds of wood gives 0.45 x 1077 = 485 pounds, close enough to 500.

The Harry Potter book weighs about 2 1/2 pounds, so one tree yields 500 / 2.5 = 200 books per tree, and

                   3,800,000 books
    total trees = ------------------ = 19,000 trees
                  200 books per tree
If the trees are growing in a square grid, then there are the square root of 19,000 = 138 trees on a side. With 10 foot spacing between the trees, this is a square 1380 feet on a side. A quarter mile is 1/4 of 5280 feet = 1320 feet, so this forest is a bit over 1/4 mile on a side. A square mile is 640 acres, so that's about a quarter of a quarter, a bit over 40 acres.
                   1380 ft x 1380 ft          acres
    Forest area =  ----------------- x 640 ----------- = 43.7 acres
                   5280 ft x 5280 ft       square mile
By sheer coincidence, that's about the size of the Dunn School campus. Forty acres and a mule, and a Quidditch field to boot!

Estimate #2: figure by volume

I already figured that an average tree has a volume of 68,000 cubic inches. The volume of the Harry Potter book is:

    volume = length x width x height
    (book) = 9.25 inch x 6.25 inch x 2 inch
           = 116 cubic inches
Therefore there are
    68,000 inch3 per tree 
    --------------------- = 586 books per tree
     116 inch3 per book
If only 45% of this volume is usable pulp, then this comes to 586 / 2 = 293 books per tree, and
                   3,800,000 books
    total trees = ------------------ = 13,000 trees
                  293 books per tree
That's fewer trees than estimate #1, but in the same ballpark.

Estimate #3: figure by industry average

The Technical Association of the Pulp and Paper Industry (TAPPI) gives this estimate at their Paper University site:

But let's assume that the following paper products have been produced using 100 percent hardwood. A cord of wood is approximately 8 feet wide, 4 feet deep, and 4 feet high. A cord of air-dried, dense hardwood (oak, hickory, etc.) weighs roughly 2 tons, about 15-20 percent of which is water. It has been estimated that one cord of this wood will yield one of these approximate quantities of products:

How many of my "average trees" are there in a cord of wood?

    cord of wood = 4 ft x 4 ft x 8 ft
                 = 128 cubic feet
                                     in      in      in
                 = 128 cubic ft x 12 -- x 12 -- x 12 --
                                     ft      ft      ft
                 = 221,184 cubic inches
We figured that a tree is 68,000 cubic inches, so that comes to 221,184 / 68,000 = 3 1/4 trees per cord.

This produces 1000 - 2000 pounds of paper, and Harry Potter weighs about 2 1/2 pounds, so

                    1000 pounds paper per cord       book     1 cord         book
    low estimate  = -------------------------- = 400 ---- x  --------- = 123 ----
                        2.5 pounds per book          cord    3.25 tree       tree

                    2000 pounds paper per cord       book     1 cord         book
    high estimate = -------------------------- = 800 ---- x  --------- = 246 ----
                        2.5 pounds per book          cord    3.25 tree       tree
This gives a total estimate of 15,000 to 31,000 trees.

That same cord of wood produces 942 100-page hard-cover books. Harry Potter is a big book with 734 pages, or 367 sheets of paper, about 3 1/2 100-page books, so that's about 942 / 3.5 = 270 books per tree. That gives another total estimate of 14,000 trees.

Estimate #4: student estimates

I don't usually assign my stumpers at school as homework, but I did assign this one because I believe that estimating (as opposed to guessing) is an important skill, and it was also a good excersize for Web searching in the new DMS computer lab. The kids used various assumptions. Unfortunately there was some convergence in our answers since many of us (me too) managed to find the same particular web sites at TAPPI, The Wilderness Society, Fall Line Consultants, and especially How Stuff Works that gives the estimate that one tree yield 805 pounds of paper. At 2.5 pounds per book, a tree yields 805 / 2.5 = 322 books per tree, and 3,800,00 / 322 = 11,801 trees, by far the most common student answer. That's easy, but I'm not happy that these kids didn't make their own assumptions. Other student answers ranged from 700 trees to 80,500 trees because of different assumptions and math errors. My estimate of 19,000 trees is on the high end. Here's a histogram of our answers:

Second Thoughts:

At the bottom of my original stumper, I said there might be something fishy about this question. DMS parent C.J. Jackson had that same thought:

In Business School,... I took three classes from a fellow by the name of Peter Drucker... One day while lecturing on incorporating the disparate elements of the management disciplines within a single case study, he asked the class what the primary ingredient of paper was. As expected, the class, not wanting to pass up a chance to shine for the Professor, raised their hands and chimed in a chorus of wood!!! Which answer, in characteristic fashion, he dismissed with the heaviest of Austrian accents, "Noahww", the primary ingredient of paper is chemicals, paper from wood had long ago been replaced by a chemical bonding process of cellulose, a portion of which may have been vegetal in origin, even from trees. Like many, I adhere to this memory, probably more for the unique characterization of the Professor, and it has wended its way into a personal gospel... The question now haunts me, was he right?

Paper is a matted felt of cellulose fiber, but those fibers comes from many sources besides trees cut just for that purpose. Scrap logs and sawdust have become too valuable to think of as waste! Especially here on the West Coast, wood chips for the pulp mills are an economically vital by-product of the timber industry, used for chemical production, and chipboard and other composites, as well as paper. This is significant. Chipboard has replaced plywood for home construction. That really does save trees!

Recycling accounts for much fiber, especially the pre-consumer recycled paper that consists of end-rolls and waste and trimmings from the paper and printing industries. The post-consumer recycled paper consists of all those old newspapers and scribbled-on envelopes and paper towels soaked in who-knows-what and all the junk printouts it takes me to get a document "just right". Paperless office? No way!

All that paper scrap we dutifully recycle at home is harder to use, along with all those old milk cartons and orange juice containers that usually don't make it to the recycle bin. I think I trust technology to find better uses for this resource that usually ends up in landfills, often dumped from the recycling yard because there's no market yet. I think there will be a market for all paper waste, especially as landfills fill like they are in Santa Barbara, and communities develop and enforce quotas. There's the ecological high-line that I usually support, but the economic bottom-line is really helping us to use the whole tree. There are alternative sources of fiber, especially Hemp aka Marijuana, if the social issues can be worked out. Synthetic fibers and "electronic ink" may change everything. But the great thing about trees is they just keep growing, and provide habitat and beauty as well!

An industry source explains it like this:

Your question - a very popular one - is not as simple as it seems. It's very difficult to calculate this figure since the fibre for paper comes from so many different sources. You could do an interesting project on the different materials that can be used to make a sheet of paper. The quality and appearance of a sheet of paper can vary enormously depending on the fibre in that sheet. Even if only wood fibre is used, there still are questions about what species of tree, what size of tree, etc.. So, we no longer try to give an answer to the question, "how many sheets of paper can come from one tree?".

Consider our school estimates an upper bound. The real answer to "how many trees does it take to print a book" may well be none! At least, that should be our goal.

There's lots of info about paper and trees on the Web that you can find by searching:

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