Analysis of regional variation of height growth and slenderness in populations of six urban tree species using a quantile regression approach

Abstract

Inventory data of six urban tree species from seven cities across Germany were used to model regional variation of height growth, allometric scaling, and slenderness over a wide range of size and age using a quantile regression approach. Variation within and between species and cities was large. Height did not reach an asymptote but declined at higher ages, presumably because of reduction cuts. Allometric scaling and slenderness varied with wind climate, modulus of elasticity, and coefficient of drag. Our data do not support the use of a threshold value of slenderness in tree risk assessment.

Introduction

Information on the height of urban trees is essential for planning tree care operations, estimating carbon stored in urban tree populations and effects of trees on micro-climate, air pollution mitigation or energy consumption (McPherson and Peper, 2012), but is often not included in municipal tree inventories. Unlike in forestry, models for urban tree growth are rare (e.g. Peper et al., 2001, Semenzato et al., 2011, Larsen and Kristoffersen, 2002). In most cases they have been developed for one or few species in one town or small regions, and for limited age and size ranges. For many of the most important tree species in central Europe, height and diameter growth and their relationships, especially at higher ages, have not been studied in urban areas.

Models parametrized in forests cannot be transferred to urban tree populations (McHale et al., 2009), because growth of urban trees differs from that of rural conspecifics (Quigley, 2004). These standard growth models from forestry might even be unsuitable to fit to urban tree populations in general, because management practices like pruning and topping introduce high variation and a decline of height with age. None of the usual models seems to be universally applicable to urban tree populations (McPherson and Peper, 2012). The lifespan of urban trees, even when not pruned, might be too short (Nowak et al., 1990) to reach the asymptote that is part of most of these models.

The ratio of tree height to stem diameter describes trunk slenderness and has been used as an indicator of stand stability in forestry for many years. Recently, this ratio has attained special importance in urban forestry because a slenderness ratio of 50 has been proposed as a failure criterion in hazard tree management (Mattheck, 2002). Since its publication, a controversy about the scientific evidence substantiating it continues (Gruber, 2008, Rust et al., 2011, Schulz, 2005, Fink, 2009, Mattheck and Bethge, 2008). But so far, there are only very few scientific studies on the development and distribution of slenderness in urban tree populations (Rust et al., 2011).

Based on mechanical theory, tree height should scale as the 2/3 power of trunk diameter (McMahon, 1973). Related research to date has tended to focus on forest trees rather than urban trees. In forest ecosystems, the height–diameter relationship of trees is influenced by factors like wood density (Ducey, 2012), maximum height (Aiba and Nakashizuka, 2009), light availability (Harja et al., 2012), and wind (Meng et al., 2008, Watt and Kirschbaum, 2011, King, 1986). The objectives of this research are to determine whether wind climate, coefficient of drag, and modulus of elasticity change allometric scaling in urban tree populations.

This paper will focus on three different approaches to model urban tree growth and its regional variation in six of the most abundant street tree species from seven cities across Germany based on inventory data of 24 599 trees up to 145 years old (Maiwald, 2012):

• non-linear quantile regression based on a sigmoid growth curve with asymptotic height to capture “ideal growth”,

• generalized additive mixed models (GAMM) to describe the average development,

• non-parametric quantile regression.

We hypothesize, that if slenderness were critical for the stability of urban trees, there should be an effect of wind climate and wood strength on tree stature. Slenderness and the proportion of large trees beyond the proposed threshold should be lower in windier cities. Based on our data, we will critically examine the validity of a single value of slenderness as a failure criterion for street trees.