In order to determine whether IAA and IBA could promote Yunnan pine seedling growth, one-year-old seedlings were treated with solutions of different concentrations of IAA in combination with IBA. Based on different experimental combinations, several regression equations were built for each growth index. One optimum regression equation was finally chosen according to correlation coefficient (R), significance level and residual standard deviation. Table 1 shows that there exists a significant correlation (P < 0.05) between each growth index and the concentration of the IAA and IBA relationship can be described by a quadric surface. There also exists a significant interaction between these two exogenous hormones in the growth of Yunnan pine.

Figure 1 shows that the effect of IAA and IBA on growth index appeared as a bivariate normal or quadratic parabola surface; that is, each growth index had one maximum peak value. Prior to the peak value, each growth index demonstrated an increase when the concentrations of IAA and IBA were likewise increased; however, after the peak value, each growth index decreased when the concentrations of IAA and IBA were increased. It can be concluded that the growth response and biomass accumulation of Yunnan pine seedlings, in conjunction with the concentration of IAA and IBA, demonstrated conformity with the law of tolerance [25–27]. An appropriate concentration of IAA and IBA could promote seedling growth, while an excessive concentration would inhibit growth. Such dual effects were reported in other investigations. Tong et al. [28] found, compared to the control alone, the addition of low concentrations of Fe-chlorophyllin resulted in an increase in the length of wheat roots. However, higher concentrations of Fe-chlorophyllin contributed to inhibition of the growth of wheat roots. Ahmed et al. [11] also confirmed that higher concentrations of IAA and IBA inhibited growth. Low and moderate concentrations of growth regulators proved to be most effective [11,29]. In Arabidopsis thaliana, Rahman et al. [30] reported that application of 10 mM auxin reached the optimum level and 100 mM greatly inhibited the elongation of Arabidopsis roots. Ivanchenko et al. [31] also verified that lateral root initiation by IAA was reduced as the IAA concentration was increased in the nanomolar range, and IAA became inhibitory at 25 nM. Similar dual effects were reported in other investigations [32–34]. This observation may be explained by the fact that an organism’s tolerance to unfavorable conditions is reduced when other environmental factors are also at non-optimal levels [35,36].

Based on the response of the auxin effect, the quadratic parabola surface vertices were the peak values of increment and biomass in the seedlings, and the corresponding optimum concentrations and ratios can be obtained. The medium concentration of IAA and IBA contributed to height growth and biomass accumulation (root, stem and needle), while high concentrations of IAA and medium concentrations of IBA contributed to the seedlings’ collar diameter growth. Figure 1 shows the effects of auxin on seedling growth at optimum concentrations and ratios. An increase in hormone levels did not lead to increased growth or biomass of the seedlings, and there exists some discrepancy in the optimum concentrations of exogenous auxin and their ratios for different growth indexes.

To gain the optimum concentrations and ratio of IAA and IBA, the same-height points (yields) were connected on each response surface and an equal yield line was gained. Vertical projections of these equal yield lines show graphs that correspond to the response surface. As Figure 2 shows, each equal yield line was distributed in concentric circles, and the circle’s central point showed the maximum yield corresponding to the optimum concentration of hormones and their ratios. Any two points on each equal yield line had the same yields, although they represented a different concentration of IAA and IBA. Therefore, we can find the single point on the same equal yield line that had the minimum concentration of IAA and IBA or the least investment for the same yield. This point represents the most economic concentration of IAA and IBA. The area covered by the Crest line OA and OB and coordinate axis was the rational concentration hormone range (Figure 2, Table 5), and the concentration of IAA and IBA beyond this area would not increase the yield but rather the investment. The OP line was the optimum concentration and ratio of IAA and IBA under the hormone price limit. Taking increments of collar diameter as an example (Figure 2b and Table 5), the rational concentration of IAA and IBA were 0~352.11 mg·L⁻¹ and 0~261.78 mg·L⁻¹, respectively. Therefore, the optimum concentrations of IAA and IBA, and their optimum ratios, can be gained from the OP line equation.

The seeds used in this experiment were harvested from a superior open-pollinate mature tree, Pinus yunnanensis, from Clonal Seed Orchard in Midu, Dali State, Yunnan Provence, China (latitude: 25°27′N; longitude: 100°28′; elevation: 1900 m). The 1000 kernel weight of seed, field germination energy and germination rate was 16.99 g, 50% and 90%, respectively. The seeds were sterilized with 0.1% HgCl₂, rinsed several times with distilled water, soaked in distilled water for 12 h and placed in a plastic film seedling container (13 cm × 17 cm) filled with soil substrate at the Lufeng Village Forest Farm of Yiliang County in Kunming, Yunnan, China (latitude: 24°30′N; longitude: 103°08′; elevation: 1800 m). There was one seed in each container. When the seedlings were two months old, they were selected for uniformity with a height and diameter of 2.85 ± 0.10 cm and 0.12 ± 0.01 cm, respectively. They were pooled and randomly distributed in each plot.

The exogenous auxins IAA (indole-3-acetic acid) and IBA (indole-3-butyric acid) were purchased from Huishi Biochemical Ltd., Shanghai, China. They are analytical reagent (AR) grade. IAA was a white powder, with more than 98% purity; IBA was light grey, with more than 99% purity. The molecular weights were 175.19 and 203.24 Da, respectively. They were dissolved in a little alcohol, diluted with distilled water and brought to a solution of 1 g L⁻¹. The different treatment solutions (Table 6) were obtained through mixing IAA, IBA solution, and distilled water.

The experiment used a 3 × 3 regression design with 2 factors and 3 levels. The concentration of IAA and IBA were selected according to the Sen B.R. [39] method, and three concentration levels of high, medium and low were chosen. Level High was two times the Level Medium concentration, and the concentration in Level Low was 0 (as control). The concentration of both IAA and IBA used in Level High was 400 mg·L⁻¹. The experiment scheme is shown in Table 6.

The experiment’s field arrangement has been optimized based on the characteristics of a Latin square. The experiment had three replications. Each replication consisted of one block composed of 9 plots (for details, see Table 7). The complete experiment contained 27 plots, with 30 seedlings per plot (n = 810 seedlings: 30 seedlings × 9 plots × 3 replicates). When seedlings were two-months-old, an IAA and IBA water solution was sprayed on the needles with a microsprayer until the needles were hung over solution but not dropped. Foliar applications were performed once every 5 days and repeated five times.

Increments were measured for every seedling, and was performed after seedlings ended their current-year growth at a time when the seedlings were one-year-old.

Biomass was measured by using a mean sample tree [40,41] method, in which the biomass of root, stem and needle of one mean sample tree was collected from each plot after the seedling ended its current-year growth, and was measured in the lab, by using the thermostatic drying oven method.

Based on the regression analysis of experimental data, a hormone effect model could be built:

where, Y is growth index; a₀, a₁, a₂, a₃, a₄, and a₅ are coefficients to be determined; A is concentration of IAA, B is concentration of IBA, AB is interaction between IAA and IBA.

From this equation, we can analyze one factor and interaction effect by using the method of descent [42], and find the optimum hormone concentration and their ratio, range of appropriate hormone concentration, and minimum cost line.