David Stockwell has suggested a discussion of nonlinear responses of tree growth to temperature. I’ve summarized here some observations which I’ve seen about bristlecones, limber pine, cedars and spruce – all showing an upside-down U-shaped response to temperature. The implications of this type of relationship for the multiproxy project of attempting to reconstruct past temperatures by assuming linear relationships between ring widths and temperature are obvious.
The assumption of a linear response between proxies (especially ring widths) is stated clearly in MBH98 as follows:
Implicit in our approach are at least three fundamental assumptions. (1) The indicators in our multiproxy trainee network are linearly related to one or more of the instrumental training patterns. In the relatively unlikely event that a proxy indicator represents a truly local climate phenomenon which is uncorrelated with larger scale climate variations, or represents a highly nonlinear response to climate variations, this assumption will not be satisfied.
This assumption is asserted in MBH98, rather than proven. Here’s what some source literature says:
Bristlecones and Limber Pine
Schoettle  Figure 4 shows an upside-down U relationship between temperature and conifer seedling growth for three species, including Great Basin bristlecone pine, limber pine and balsam fir.
Original Caption: Figure 4“¢’¬?Relative temperature response of net photosynthesis of seedlings of three conifer species. The optimum temperature for photosynthesis for each species is that temperature that the maximum rate of photosynthesis was recorded. To enable comparison among species, photosynthesis is expressed as a percentage reduction from the maximum rate.
Schoettle observed that there were sharp declines in bristlecone net photosynthesis with increasing temperature (and a lesser decline in limber pine):
How can limber pine uncouple its growth from the environmental differences from the upper to the lower tree line? The rates of most physiological and biochemical processes are a function of temperature. Limber pine seedlings from four of five populations from Wyoming, Nevada and California revealed a typical photosynthetic temperature optimum (15 à⣃ ’ à ⽃) but an unusually broad response curve with a variation in photosynthetic rate of only 12 percent from the maximum over the temperature range of 10-35 deg C (Lepper 1980). This is in contrast to the sharper temperature response of photosynthesis of balsam fir (Abies balsamea (L.) Mill., Fryer and Ledig 1972) and Great Basin bristlecone pine (then called Pinus aristata Engelm. but now recognized as Pinus longaeva Bailey) according to Bailey (1970) Mooney and others (1964) where photosynthesis fell 63 percent and 87 percent, respectively, below the maximum rates within the range of 5 deg C below and 20 deg C above the optimum temperature for photosynthesis (fig. 4). Strong variation in photosynthetic capacity between mature trees at the elevational extremes (Schoettle, unpublished data) also suggests considerable adaptive physiological variation for limber pine.
Some other interesting observations in Schoettle  (not exactly on point but collated here since I found the points interesting):
It is thought that during the Pleistocene glacial periods there was nearly continuous habitat for bristlecone pine between the New Mexico and Arizona stands, suggesting that the Arizona stand is a relic of a formerly larger distribution (Bailey 1970). The current southern distribution of bristlecone pine appears limited by suitable habitat, however it is not known what limits bristlecone pine from occupying apparently suitable habitat to its north. The distribution of this species may reflect a dependence on summer monsoons, restricting it from occupying higher elevation sites in northern Colorado. Rocky Mountain bristlecone pine (referred to as bristlecone pine hereafter) has a narrow elevation range and is primarily a high elevation species occupying dry sites from 2750 to 3670 m elevation (Baker 1992)…
The origin of bristlecone pine stands throughout Colorado is related to episodes of drought and presumably peak fire occurrence (Baker 1992). Bristlecone pine is a long-lived species that regenerates well after fires. Baker (1992) reports that bristlecone pine regenerates well only on recently burned sites and therefore attributes the persistence of old stands of bristlecone not to climax stand dynamics but to the long lifespan of the individual pioneer trees in the absence of competition and fire. However, Baker’s data reveal some bristlecone pine regeneration in most of the sampled bristlecone pine stands. This raises the question of how much regeneration is necessary to sustain bristlecone pine on sites with little to no competition…
Vegetation in bristlecone forests is influenced primarily by elevation and soil pH and secondarily by substrate, soil texture, topographic position, and geographic location (Ranne and others 1997). Although bristlecone pine is a pioneer species after fire, its role in mediating the environment to facilitate the establishment of late successional species has not been fully explored. In the subalpine zone, bristlecone pine forests tend to have relatively clear boundaries with bristlecone pine densities abruptly falling as elevation decreases and moisture regimes change.
Larson, Kelly and Matthes of the University of Guelph have been studying cedars for over 15 years with remarkably interesting results. The cedars in question occur in cliffs along the Niagara Escarpment which stretches through southern Ontario from Niagara to the Bruce Peninsula. The "forest" is about 600 km long and 100 meters wide. They have pointed out many similarities between these cedars and bristlecones – both of which have strip bark forms, are very long-lived and live in very adverse environments. Kelly et al  (including Hockey Team member Ed Cook) reported an "optimum" temperature above and below which growth decreases:
The strong negative correlation between tree growth and mean monthly temperature shows that tree ring growth is specifically inhibited by hot July and August maximum temperatures in the preceding summer. T. occidentalis may respond to excessive late summer temperatures by slowing down physiologically in a way that influences growth potential in the following year. This agrees with observations made by Matthes-Sears and Larson  on the net photosynthesis of T occidentalis in response to changing leaf temperatures. In that study, net photosynthetic rates in Thuja were shown to peak at 20 degree C and decline with continued increases in temperature leading to markedly reduced rates of photosynthesis beyond 30 deg C. The exact physiological mechanisms in T occidentalis that explain this response to high temperatures are not know, but they likely involve increases respiration, lower net photosynthesis [Fritts, 1966, 1976] and the resultant loss of photosynthetic reserves.
A few tree ring studies indicate recent growth declines at northern latitudes. The precise causes are not well understood. Here we identify a temperature threshold for decline in a tree ring record from a well-established temperature-sensitive site at elevational tree line in northwestern Canada. The positive ring width/temperature relationship has weakened such that a pre-1965 linear model systematically overpredicts tree ring widths from 1965 to 1999. A nonlinear model shows an inverted U-shaped relationship between this chronology and summer temperatures, with an optimal July–August average temperature of 11.3 deg C based on a nearby station. This optimal value has been consistently exceeded since the 1960s, and the concurrent decline demonstrates that even at tree line, trees can be negatively affected when temperatures warm beyond a physiological threshold.
D’Arrigo et al.  go on to say:
We chose TTHH for this case study because it is a well-established temperature sensitive site, where the data for ring width (including raw measurements as well as standardized indices) show decline since the 1960s despite a concurrent rise in Arctic temperature [Chapman and Walsh, 1993; Hansen et al., 1999]. At nearby Dawson, Yukon Territory (see section 2), temperature has increased along with the Arctic warming (see Figure 4), while precipitation has not changed significantly. These observations suggest that persistent higher temperatures may have induced a stress on the trees. The stress may be due to temperatures rising above an optimum level for growth at this site. Above a certain level, net photosynthesis declines as the effects of thermally increasing respiration overcome the diminishing response of photosynthesis to temperature increases [Kramer and Kozlowski, 1979].
Original Caption: Figure 3. Results of a linear relationship between temperature and tree rings. Actual (solid line) tree ring width indices for TTHH chronology are shown. Estimates (dashed line) based on using Dawson temperatures as predictors over 1901–1964 period are shown. (These temperatures are based on adjusted data from Vose et al. . Use of adjusted data was recommended by D. Easterling (Scientific Services Division, National Oceanic and Atmospheric Administration’s National Climatic Data Center) and L. Vincent (Canadian Meteorological Service) (personal communication, 2004).) Horizontal line is tree ring index mean over 1901–1999. Variance accounted for by linear regression model is 33% (ar2), adjusted for degrees of freedom. Model is based on temperature variables: Year t-1 is April, May, July, and August; year t is May–August. Note that regression estimates for 1965–1999 overpredict ring width values. Analyses using unadjusted data indicated some differences relative to those reported herein (with less of a disparity between actual and estimated tree growth) but the same overall conclusion: i.e., that radial growth at this site is overpredicted by the linear model based on temperature.
Original Caption Figure 4. Comparison of observed July–August averaged temperature values (thick line) with physiological optima (straight thin line) computed for TTHH site. Because there are relatively few years in which observed temperature exceeds the threshold prior to 1965, the correlation between ring width and temperature is generally positive (Figure 2).
D’Arrigo et al  go on to say:
As noted in text, elevational differences between Dawson and the TTHH site imply that summer temperatures could be 3.5 deg C lower at the tree site [Jacoby and Cook, 1981].Our results indicate that the tree growth decline at TTHH is consistent with increases in the average July–August temperature over its optimal value (Figure 4). Unlike the systematic error in the linear model (Figure 3), which highlights its lack of cointegration (ADF with constant and no time trend is 2.25, p > 0.38), cointegration implies that there is no systematic error in the quadratic model. The lack of a systematic error suggests that higher July–August temperatures (beyond the trees’ physiological optima) are consistent with the growth decline at TTHH after 1965.These results, combined with those for the ECM, suggest a mechanism for the quadratic relationship between ring width and July–August temperature. Early in the growing season, soil moisture can be sufficient for growth even at somewhat higher temperatures. For example, the coefficient associated with June temperature in equation (3) is positive. As summer progresses, soil moisture usually dwindles. The lack of moisture is especially stressful at high temperatures. Without a persistent increase in precipitation (the precipitation time series are stationary), the persistent increase in temperature over time beyond the 11.3 deg C threshold has a negative effect on ring width. Toward the end of the growing season when soil moisture probably is near the growing season low, the lagged values in the ECM generally have a negative effect on ring width. This would indicate that warm temperatures have a negative effect on ring width in the absence of sufficient soil moisture.
How can this direct evidence of a relationship between temperature and ring width which is not only non-linear, but non-monotonic and reversing within the very target temperature range, be reconciled with the linear assumption of MBH98 and other multiproxy studies? Beats me.
A.W. Schoettle, 2004. Ecological Roles of Five-Needle Pines in Colorado: Potential Consequences of Their Loss In: Sniezko, Richard A.; Samman, Safiya; Schlarbaum, Scott E.; Kriebel, Howard B., eds. 2004. Breeding and genetic resources of five-needle pines: growth, adaptability and pest resistance; 2001 July 23–27; Medford, OR, USA. IUFRO Working Party 2.02.15. Proceedings RMRS-P-32. Fort Collins, CO: U.S. Department of Agriculture, Forest Service, Rocky Mountain Research Station. http://www.fs.fed.us/rm/pubs/rmrs_p032/rmrs_p032_124_135.pdf
P. Kelly, E.R. Cook and D.W. Larson, 1994. A 1397 tree ring chronology of Thuja occidentalis from cliff faces of the Niagara Escarpment, southern Ontario. Cdn J For Res 24, 1049-1057.
Matthes-Sears U and D.W. Larson, 1990. Environmental controls of carbon uptake in two woody species with contrasting distributions at the face of cliffs. Cdn J Bot 68, 2371-2380.
D’Arrigo, R. D., R. K. Kaufmann, N. Davi, G. C. Jacoby, C. Laskowski, R. B. Myneni, and P. Cherubini (2004), Thresholds for warming-induced growth decline at elevational tree line in the Yukon Territory, Canada, Global Biogeochem. Cycles, 18, GB3021, doi:10.1029/2004GB002249. http://cybele.bu.edu/download/manuscripts/darrigo01.pdf