## Introduction

Nitrogen (N) availability is crucial for forest production, but N uptake imposes carbon (C) costs for fine-root construction (Davidson, 1969) and maintenance respiration (Ryan, 1991). This suggests that an optimal pattern of C and N co-allocation exists. In an evolutionary sense, nature may select for plants that optimize their use of available resources and achieve high productivity. In this regard, it has been suggested that a model of optimal co-allocation of N and C production could advance our understanding of plant responses to variable N and C supplies (Johnson & Thornley, 1987; Mäkelä & Sievänen, 1987; Hilbert, 1990; McMurtrie, 1991; Dewar, 1996; Ågren & Franklin, 2003; Franklin, 2007).

The results of optimization studies largely depend on the assumptions used to constrain the problem. A balanced-growth approach is based on the assumption that the tissue N concentration is maintained at a ‘balanced level’ by optimal allocation of substrate to shoots and roots. The assumption that allocation controls the balance essentially relies, in turn, on the assumption that shoot and root activities in C and N acquisition depend on organ size. The balanced level may be either an *a priori* assumption reflecting a demand for N defined by the stoichiometries of the shoot and root tissues (Mäkelä & Sievänen, 1987), or an additional result of the optimization problem, based on additional assumptions concerning the impacts of N on growth and production (Johnson & Thornley, 1987; Hilbert, 1990; Ågren & Franklin, 2003). Hilbert (1990) showed that, if the rate of photosynthesis was a saturating function of the N concentration in leaves, then the optimal concentration varied smoothly with N availability. Optimal allocation was a compromise between N costs incurred by C allocation to roots, and N gain from an increase in photosynthesis due to increased N concentration in leaves.

The above studies were confined to analysing unconstrained, exponential growth, defining relative growth rate as the objective function to be maximized. Other investigators have analysed optimal steady-state canopies, where leaf N concentration impacts physiological processes (McMurtrie, 1991; Dewar, 1996; Franklin & Ågren, 2002; Franklin, 2007). However, the cost of N acquisition is absent from these models. McMurtrie (1991) assumed a canopy connected to a root system sufficient for the uptake of all available N. Dewar (1996) derived an optimal canopy N content for a given leaf area index (LAI), and Franklin (2007) derived an optimal leaf area index for a given N content, but no explicit cost was attached to the construction or maintenance of the root system that would be required to acquire the assumed N content.

The cost of constructing a root system is explicit in the balanced-growth models, but the exponential-growth studies have implicitly assumed that the requirement for N can always be satisfied by suitable growth allocation to roots, regardless of the size of the plant. At the stand level, this assumption must be tempered by the reality of limited N availability.

In forest stands, the limitation of resources leads to closed, or nonexpanding, canopies. These situations can be analysed effectively by considering the C and N balances of the stand at steady state (Dewar, 1996; Franklin, 2007). Strictly speaking, the steady-state assumption does not apply to woody biomass, as stem elongation continues until stand senescence (Mäkelä & Valentine, 2001). However, the elongation growth utilizes a small fraction of net primary productivity (NPP) (Mäkelä, 1986), so, at appropriate timescales, the steady-state assumption can nevertheless serve as a realistic approximation for resource-limited stands.

The objective of this study was to explore the extent to which optimal co-allocation of C and N at steady state explains observed responses of forest traits to variable N and C supply and climate. Do optimal closed-canopy foliage density and above-ground allocation increase with N availability? How do these patterns interact with leaf N concentration and canopy height? Is light use efficiency (LUE) independent of N availability? How do optimized stands respond to increasing atmospheric CO_{2}? Is there a climate-independent relation among foliage density, fine-root density and leaf N concentration?

We consider these questions by means of a model that maximizes net production at steady state by optimizing the co-allocation of C and N to leaves, fine roots, and live wood, given the maximum uptake rate of N, and accounting for the costs of production and maintenance respiration.