Research Note: The Tripping Point – Minimum Planting Widths for Small-Stature Trees in Dense Urban Developments

As urban development increases in density, the space to grow urban trees becomes more constrained. In heavily developed areas, small stature trees can be planted to reduce both above-and below-ground conflicts with infrastructure elements. However, even these species have their limits when placed in extremely confining conditions. In this study, we build on past work to determine the minimum planting widths of small stature urban trees. We found that species, stem diameter, and the height at which stem diameter measurements occurred were all strong predictors of trunk flare diameter (adjusted R 2 of 0.843). Additionally, we modelled the relationship between planting space and the presence or absence of hardscape conflicts – using the predictions derived from this effort to project the potential cost savings in two United States cities. Study results provide a guideline to create sufficient space for urban trees and minimize infrastructure damage and associated cost savings.


Introduction
Urban trees are often integrated into urban planning and design efforts to increase the walkability of neighborhoods (Choi et al., 2016), calm traffic (Van Treese et al., 2017), and shade parking areas (Koeser et al., 2021;Grabosky and Gilman, 2004). Their ability to cool localized environments through the absorption of the sun's light energy and the transpiration of soil moisture has been used effectively to mitigate the impacts of urban heat island buildup (Petri et al., 2019) as well as to extend the life of some paving materials (McPherson and Muchnick, 2005). This noted, conflicts between tree roots and the paved surfaces that facilitate pedestrian and vehicular traffic are a common ecosystem disservice (Roman et al., 2020) and a major municipal expense (McPherson, 2000;Randrup et al., 2003).
From an engineering perspective, trees and their roots represent a significant factor contributing to the lifting, cracking, or general degradation of paved surfaces. In his assessment of the costs of tree and hardscape conflicts in 18 California (United States) communities, McPherson estimated that $114.3 million total or $4.48 per capita (USD; CPI adjusted to 2021 values) was spent annually due to tree conflicts and replacing sidewalks, curbs, and gutters across the whole of the state. In more recent conversations with local transportation engineers, tree and sidewalk conflicts are still seen as the primary cause necessitating sidewalk replacement (Gorman, City of Tampa; personal communication).
Root and hardscape conflicts can be similarly damaging to the trees involved. The replacement or repair of paved surfaces near trees can sever or injure roots -reducing tree health (Benson et al., 2019;Hauer et al, 2020;Koeser et al., 2013) and undermining overall stability in the face of storm events (Johnson et al., 2019). To reduce root and hardscape conflicts, researchers have investigated how planting widths relate to sidewalk damage (Francis et al., 1996;Randrup et al., 2003) and created allometric models to predict trunk flare (i.e., the enlarged area at the base of the tree where the trunk connects to the main structural roots) diameter based on tree species and stem diameter (Hilbert et al., 2020;North et al., 2015).
In this extension of past research by North et al. (2015) and Hilbert et al (2020), we developed allometric models linking stem diameter to trunk flare diameter in small-stature urban trees. While the large stature shade trees assessed by the two research teams cited above are important contributors of ecosystem services, modern compact development patterns leave less space for the sustained growth of large trees (Daniel et al. 2016). As cities continue to densify, small understory trees may be the best choice for urban greening efforts (North et al., 2015). This noted, even smallstature trees have their limits with regard to minimal space allotments. Given the general lack of knowledge surrounding the planting space requirements of small stature trees, we set the following research objectives for this study: 1.) Develop a set of equations that can be used to estimate root space requirements for small-stature urban tree species; and 2.) Determine the minimum allowable planting space for trees typically selected for space-limiting planting conditions. In addressing these two aims, we provide planners, policy makers, engineers, urban foresters, and others with guidelines for more informed planting and design strategies in compact urban developments.

Materials and Methods
We worked with local urban foresters to locate and measure small stature urban trees in Lakeland Trunk flare circumference was determined using the method of Hilbert et al. (2020). In brief, marking flags delineated points at the base of the tree where the root stem transition zone transitioned to root tissue. A nylon measuring tape was placed against the outside of each flagged point in a circular manner to measure the circumference which was later converted to trunk flare diameter used in the analysis described later. For each tree, the location, species, stem diameter, distance to nearest hardscape, and any hardscape damage were noted. Stem girdling roots and buried structural roots were also recorded since these might influence trunk flare diameter and tree health (Hauer and Johnson, 2021). Height of diameter measurements (Dx) were collected at one of three locations on the tree (i.e., at 137 cm, 15.25 cm, or 5 cm), since measuring at a standard height of 137 cm was not possible in all cases. If the tree was of sufficient height and pruned to elevate the crown, then diameter was measured at 137 cm. If the tree's stem split at or below 137 cm, but the stems merged above ground, then the diameter was measured at caliper height (15.25 cm). If the tree was multi-stemmed, then the diameter was recorded at the base of the tree (5 cm).
A series of multiple linear regression models were used to determine the relationship between trunk flare diameter and species, stem diameter, and the diameter measurement height. These analyses were conducted using the lm() function in R (R Core Team, 2021). Diagnostic plots (e.g., residuals versus fitted values, Q-Q plots, and residuals versus leverage) were generated to assess adherence to the underlying model assumptions and to determine if any high-leverage outliers were influencing our predictions.
In predicting sidewalk damage, we used logistic regression to determine if stem diameter, measurement height, distance to hardscape, or some combination of these main effects influenced the presence or absence of hardscape lifting or cracking. Modelling was conducted using the glm() function in R (R Core Team, 2021). Cross validation error rate was determined using the cv.glm() function from the boot package in R (Canty and Ripley, 2021). Additionally, an ROC curve and its associated AUC value were created/calculated using the ROCR package in R (Sing et al., 2005).
An α=0.05 was adopted as a threshold of statistical significance.

Results and Discussion
Species, Dx, and diameter measurement height were all significant predictors of TFD (Table 1).
This noted, the overall predictive power of a simplified model (adjusted R 2 = 0.790) where species was not included as a predictor variable was similar to the more inclusive full model (adjusted R 2 = 0.843; Table 1). Use of the former model may be preferred for simplicity or when working with species beyond those included in this study. To this point, the coefficients for Dx (full model =  (Table 1).
Of the 288 trees measured, only 33 (11.5%) were associated with damaged hardscape. Cracking was the most common damage category (n=19), followed by pavement lifting (n=10), and other (n=4). Both stem diameter (Dx) and distance from hardscape were significant predictors of the presence of damage when modelled singly, though when modelled together the latter predictor dropped out given non-significance. As distance from hardscape is the factor professionals have the greatest control over, we adopted a simple model with this as the sole variable for predicting hardscape damage (P-value < 0.001; cross-validation error rate = 6.7%; AUC = 0.742). In calculating the odds ratio from the distance from hardscape coefficient, we found damage was 1.015 times less likely to occur if planting width was increased by 1 cm. More meaningful spacing comparisons are featured in Figure 1.  reported by Hilbert et al. (2020).
To put this into perspective for urban planners, urban forest managers, and transportation engineers, we reached out to our expert contacts at the City of Tampa (Florida, United States) and the removal of the old slab, associated tree work, and the pouring of the replacement slab (Gorman, personal communication). Projected savings are found in Table 3 below.
The City of Milwaukee has a sidewalk replacement budget of $1.5 million (USD) annually (Kringer, personal communication). Sidewalk replacement is charged as an assessed fee to the associated homeowner. The expense of replacement is partially offset by a citywide tax on motor vehicle registrations. To replace the same ~1.5 m wide by ~1.5 m long slab noted above, homeowners would be assessed a $95 (USD) fee (actual contracted costs were not available to our contact). As such, the savings calculated for the Milwaukee scenario (Table 3) are savings to the associated homeowner and not the City itself.  Table 3 shows the estimated per tree savings in sidewalk replacement costs as the distance between the base of a tree and neighboring hardscape is increased. In Tampa, increasing distance to hardscape from 0 cm to 100 cm would save approximately $120 (USD) per tree. It would take 200 cm to achieve a similar savings for large stature trees in the city.
Finally, in applying this research to practice, we suggest the following equation for determining minimum planting width (modified from Hilbert et al. 2020 andNorth et al 2015): Where: = minimum planting width = predicted trunk flare diameter Predicted trunk flare diameter may be calculated using either the full or simplified model (Table   1) -drawing on existing urban forest inventory data to determine the growth potential of a tree species in one's local urban environment. This is divided by 100 to convert cm to m and a 1.2 m buffer is added to account belowground structural roots. This buffer is halved from the large tree equations proposed by Hilbert et al. (2020) and North et al. (2015) given the reduced potential for damage noted in Figure 1.