Abstract
Because forests take up carbon dioxide, “protecting forests” and “growing forests” are easily taken to point the same way. What this paper shows is that the rate at which a forest takes up carbon, the mean rate, and the amount it holds each peak at a different age: young forests take up, old forests hold. No new theorem or law is claimed.
First, the rate of uptake peaks at $\mathbf{27.465307}$ years——the closed form $\ln3/k$ and the numerical peak agree, giving $\mathbf{8.8889}$ m$^3$/ha/yr, or $\mathbf{8.4741}$ t of carbon dioxide per ha per year (Section 2).
Second, the mean rate of uptake peaks at $\mathbf{47.595342}$ years——there the rate and its mean agree at $\mathbf{6.474337}$ m$^3$/ha/yr, or $\mathbf{6.1722}$ t of carbon dioxide per ha per year (Section 3).
Third, the stock keeps growing——$\mathbf{0.975513}$ of the ceiling at $120$ years, $\mathbf{5.841932}$ times a $20$-year-old forest, while the rate of uptake is only $\mathbf{0.054640}$ of its peak (Section 4).
Fourth, and this is the core. Three peaks lie at three ages——the rate a little over $27$ years, the mean rate just under $48$, and the stock endlessly later. Young forests take up; old forests hold (Section 4).
Fifth, the peak of the mean is flat——felling at $40$ years still gives $\mathbf{0.981505}$ of it, and at $55$ years $\mathbf{0.987201}$; yet at $60$ years the stock is $\mathbf{0.751789}$ of the ceiling and the rate $\mathbf{0.506284}$ of its peak, both moving a great deal (Section 5).
Sixth, the separator is flow against stock——felling and replanting at the peak of the mean maximizes the average rate of uptake, but drops the average stock to $\mathbf{0.254899}$ of the ceiling (Section 5).
Keywords: forest carbon, Chapman-Richards growth curve, current annual increment, mean annual increment, stand age.
Scope (what is not claimed)
No new theorem or law is claimed——the Chapman--Richards growth curve and the relation between current and mean annual increment are standard. The growth curve is representative——a ceiling of $500$ m$^3$/ha, a rate constant of $0.04$ per year and an exponent of $3$; species and site change these greatly. The conversion to carbon is representative——wood density $0.4$ t/m$^3$, a factor of $1.3$ for branches, leaves and roots, and a carbon fraction of $0.5$. Carbon in harvested wood is not included——if felled timber lasts long in houses and furniture, its carbon stays held; this paper counts only the stock in the forest. Soil carbon, mortality, thinning and disturbance are not included. No forestry policy is recommended——“fell or protect” is not answered; what is shown is that the answer changes with what is maximized. Relation to earlier papers: Paper 547 showed that the optimum order quantity is flat around itself, so it need not be held exactly——the peak of mean increment is flat in the same way; this paper sets beside it the large movement of other quantities near that flat one. Paper 386 showed that “hard to dissolve” points at two quantities, solubility and rate of dissolution——“a forest that takes up well” also points at two, rate and stock. Paper 300 showed that sharing a root is decidable——the three peaks come from one growth curve and share a root, but measure different quantities. What is added is confirming the peak of the rate against the closed form $\ln3/k$ numerically, confirming that the rate equals its mean at the peak of the mean, setting out how the rate falls while the stock goes on growing, obtaining the average stock over a rotation by a closed form and by numerical integration, and placing the separator on the distinction between flow and stock.
1. Notation and the Question
Setting: the volume of a stand (the stock) is written with the Chapman--Richards growth curve
$$V(t)=V_{\max}\left(1-e^{-kt}\right)^{3}.$$The rate of uptake is the current annual increment $V'(t)$, and the mean rate over a rotation of felling and replanting is the mean annual increment $V(t)/t$. Volume is converted to carbon dioxide by wood density, a factor for branches, leaves and roots, the carbon fraction and $44/12$.
Observation 1.1 forests take up carbon dioxide. So “protecting forests” and “growing forests” are easily taken to point the same way——but a forest that takes up well and a forest that holds much need not be the same forest.
Question: at what ages do the rate of uptake, the mean rate and the stock each peak?
2. The Peak of the Rate
Computation 2.1: with a ceiling of $500$ m$^3$/ha and $k=0.04$ per year, the peak of $V'(t)$. The closed form $\ln3/k=\mathbf{27.465307}$ years agrees with the numerically found peak to $10^{-4}$ years.
Observation 2.2 the rate of uptake is greatest in a young forest. At the peak it is $\mathbf{8.8889}$ m$^3$/ha/yr, or $\mathbf{8.4741}$ t of carbon dioxide per ha per year——a forest under thirty years old, not yet at half its ceiling, takes up fastest.
3. The Peak of the Mean Rate
Computation 3.1: the peak of the mean increment $V(t)/t$, found by solving $V'(t) t=V(t)$. It lies at $\mathbf{47.595342}$ years, where the rate and its mean agree at $\mathbf{6.474337}$ m$^3$/ha/yr (to $10^{-9}$).
Observation 3.2 the peak of the mean comes twenty years after the peak of the rate. It is $\mathbf{6.1722}$ t of carbon dioxide per ha per year——if felling and replanting are repeated, felling at this age takes up the most per year on average.
4. The Stock
Computation 4.1: stock and rate of uptake by age.
| age | stock (m$^3$/ha) | of the ceiling | rate (m$^3$/ha/yr) | of the peak |
|---|---|---|---|---|
| $20$ yr | $\mathbf{83.4924}$ | $\mathbf{0.166985}$ | $\mathbf{8.1752}$ | $\mathbf{0.919714}$ |
| $40$ yr | $\mathbf{254.1837}$ | $\mathbf{0.508367}$ | $\mathbf{7.7161}$ | $\mathbf{0.868063}$ |
| $60$ yr | $\mathbf{375.8944}$ | $\mathbf{0.751789}$ | $\mathbf{4.5003}$ | $\mathbf{0.506284}$ |
| $80$ yr | $\mathbf{441.3152}$ | $\mathbf{0.882630}$ | $\mathbf{2.2504}$ | $\mathbf{0.253171}$ |
| $120$ yr | $\mathbf{487.7567}$ | $\mathbf{0.975513}$ | $\mathbf{0.4857}$ | $\mathbf{0.054640}$ |
| $200$ yr | $\mathbf{499.4970}$ | $\mathbf{0.998994}$ | $\mathbf{0.0201}$ | $\mathbf{0.002263}$ |
Observation 4.2 this is the core. The rate peaks a little after $27$ years, the mean rate just before $48$, and the stock endlessly later——three peaks at three ages. Young forests take up; old forests hold.
Observation 4.3 an old forest takes up almost nothing and holds a great deal. A $120$-year-old forest holds $\mathbf{5.841932}$ times as much as a $20$-year-old one, yet takes up at only $0.054640$ of the peak rate——neither “a forest that no longer takes up is useless” nor “a forest that holds most is best” follows; the quantities measured differ.
5. Felling and Replanting
Computation 5.1: the average stock over a rotation felled and replanted at the peak of the mean, $47.595342$ years. The closed form and numerical integration agree to $10^{-6}$: $\mathbf{127.4494}$ m$^3$/ha, or $\mathbf{0.254899}$ of the ceiling. The mean rate of uptake over that rotation is $\mathbf{6.4743}$ m$^3$/ha/yr.
Observation 5.2 maximizing the mean rate cuts the mean stock to a quarter. An aged forest takes up almost nothing but holds close to its ceiling——felling and replanting keeps the rate up, but what can be held in the forest falls to about a quarter of the ceiling.
Computation 5.3: the felling age moved either side of the peak of the mean.
| felling age | mean rate (of the peak) |
|---|---|
| $35$ yr | $\mathbf{0.943602}$ |
| $40$ yr | $\mathbf{0.981505}$ |
| $55$ yr | $\mathbf{0.987201}$ |
| $60$ yr | $\mathbf{0.967652}$ |
Observation 5.4 the peak of the mean is flat. Moving the felling age from $40$ to $55$ years never drops the mean rate below $98$ per cent of its peak——yet at $60$ years the stock is $0.751789$ of the ceiling and the current rate $0.506284$ of its peak: other quantities move a great deal. Watching only the flat one hides those that move.
6. The Separator
Observation 6.1 the separator is flow against stock. The rate of uptake is the slope of the growth curve (a flow); the stock is its height——the flow peaks partway along the curve, the stock grows to the end. Which is maximized changes the answer to the best age.
Observation 6.2 flat around the optimum, as in Paper 547. There, doubling the order quantity raised the total cost only to $1.25$ times——here, shifting the felling age hardly lowers the mean rate. What this paper adds is that beside that flat quantity, the stock and the current rate move a great deal.
Observation 6.3 the shape of Paper 386. There, “hard to dissolve” pointed at two quantities, solubility and rate of dissolution——here, “a forest that takes up well” points at two, rate and stock.
Observation 6.4 one root under Paper 300, different quantities. All three peaks come from one growth curve——what differs is whether the slope, the mean of the slope, or the height is looked at.
Observation 6.5 what is not said. No forestry policy is recommended——with carbon in harvested wood left out, “fell or protect” cannot be answered here. What is shown is that the answer changes with what is maximized.
7. Roll Call and Closing
- the rate peaks at $\mathbf{27.465307}$ years (Computation 2.1): matching $\ln3/k$.
- $\mathbf{8.4741}$ t CO$_2$/ha/yr at the peak (Observation 2.2): young forests are fastest.
- the mean peaks at $\mathbf{47.595342}$ years (Computation 3.1): rate and mean agree at $\mathbf{6.474337}$.
- $\mathbf{0.975513}$ of the ceiling at $120$ years (Computation 4.1): the stock keeps growing.
- three peaks at three ages (Observation 4.2): the core.
- $\mathbf{5.841932}$ times the stock, $\mathbf{0.054640}$ of the rate at $120$ years (Observation 4.3): different quantities.
- mean stock over a rotation $\mathbf{0.254899}$ of the ceiling (Computation 5.1): the price of keeping the rate up.
- $\mathbf{0.981505}$ at $40$ years, $\mathbf{0.987201}$ at $55$ (Computation 5.3): the peak of the mean is flat.
- the separator is flow or stock (Observation 6.1): slope or height.
- flat, as in Paper 547 (Observation 6.2): other quantities move beside it.
Closing: Because forests take up carbon dioxide, “protecting forests” and “growing forests” are easily taken to point the same way. Yet the rate of uptake, the mean rate and the stock each peak at a different age——under Chapman--Richards growth the rate peaks, as the closed form $\ln3/k$ says, at $\mathbf{27.465307}$ years, at $\mathbf{8.4741}$ t of carbon dioxide per ha per year. The mean rate peaks at $\mathbf{47.595342}$ years, where rate and mean agree at $\mathbf{6.474337}$ m$^3$/ha/yr. The stock goes on growing, to $\mathbf{0.975513}$ of its ceiling at $120$ years and $\mathbf{5.841932}$ times a $20$-year-old forest, while the rate then is only $\mathbf{0.054640}$ of its peak. Young forests take up; old forests hold——felling and replanting at the peak of the mean maximizes the average rate of uptake, but the average stock held in the forest falls to $\mathbf{0.254899}$ of the ceiling. The peak of the mean is flat, still $\mathbf{0.981505}$ at $40$ years and $\mathbf{0.987201}$ at $55$, yet at $60$ years the stock is $\mathbf{0.751789}$ of the ceiling and the rate $\mathbf{0.506284}$ of its peak, both moving a great deal. The separator is flow against stock——the rate of uptake is the slope of the growth curve and the stock its height; the flow peaks partway, the stock grows to the end. Under Paper 300 the three peaks share the root of one growth curve but measure different quantities. Placed among the earlier papers——as in Paper 547 the optimum is flat around itself, and this paper sets beside it the other quantities that move a great deal. As “hard to dissolve” pointed at two quantities in Paper 386, “a forest that takes up well” points at two. To be honest——the growth curve and the conversion to carbon are representative, and carbon in harvested wood, soil carbon, mortality, thinning and disturbance are left out. No forestry policy is recommended: what is shown is that the best age changes with what is maximized.
References
- Davis, L. S., Johnson, K. N., Bettinger, P., & Howard, T. E. (2001). Forest Management: To Sustain Ecological, Economic, and Social Values (4th ed.). McGraw-Hill.
- Pretzsch, H. (2009). Forest Dynamics, Growth and Yield. Springer.
- Richards, F. J. (1959). A Flexible Growth Function for Empirical Use. Journal of Experimental Botany.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Solubility Did Not Move at All, and the Dissolution Became a Thousand Times Faster [Paper 386]. Zenodo. 10.5281/zenodo.22699119
- Yamagishi, Y. (2026). The Optimum Is Flat Around Itself, So It Need Not Be Held Exactly [Paper 547]. Zenodo. 10.5281/zenodo.23014175
Reference
Yamagishi, Y. (2026). The Age at Which a Forest Takes Up Carbon Fastest Is Not the Age at Which It Holds the Most [Paper 581]. YOI: https://www.yamagishiyuuki.com/yoi/581
DOI: 10.5281/zenodo.23014343
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 581. This paper is a preprint and has not been peer reviewed.