Here we present a two dimensional size-structured model where both root and leaf are growing. Trees are the result not only of optimization, but of an evolutionary game. For example the stupendous trunks of trees are a result of an evolutionary game for gaining light by over topping neighbour trees. The optimal tree would not invest in a trunk since it is very costly. Likewise roots are involved in a game of increasing water uptake, with the result of an over proliferation of roots.

We assume that there is competition for light and water, and evolve the environmentally depending growth allocation to leaf and fine root, which is clearly a trait with a trade-off since trees cannot survive with only leaf or only fine root. Competition for light is asymmetric as large individuals shade small individuals. Competition for water is not so clearly asymmetric since the roots of large individuals do not totally dominate in the most upper layers of soil, where you expect to find the roots of small seedlings. Instead they invest in deeper roots to access the more secure deeper soil water.

Root and shoot allocation has been studied extensively, and much is known about the basic principles. For example root shoot ratios depend on nitrogen productivity, internal nitrogen and photosynthetic rate. However, plants typically do not use the same allocation during the whole life cycle, since the allocation depends on the environment, which changes over time due to biotic and abiotic factors. Generally the root-shoot allocation is balanced until maturation, when all resources go into reproduction and maintenance. When removing leaves or root manually, plants typically respond by putting all resources into leaf or root production until the optimal root shoot allocation is recovered. This balanced growth path is believed to optimize reproduction.

Our main questions are: 1) How does light and water availability affect leaf-root allocation? 2) Does root distribution affects leaf-root allocation? and 3) Can root mass can be derived from above-ground mass?