Tree height and crown shape, as results of competitive games

Games between trees are studied to explain the height and the crown shape in a community. It is assumed that each tree maximizes the net productivity under the light environment, which is determined by surrounding trees. In both models, the asymmetry of competition is important, because higher leaves shade lower leaves but the lower don't shade the higher. In the tree-height game, each tree is assumed to choose a height given its crown shape. The height at a monomorphic equilibrium, in which all the trees of a community have the same height, increases with the tree density and the amount of leaves per tree, but decreases with the cost coefficient and the crown thickness. When the crown is thin enough, a polymorphic equilibrium appears, in which a community includes trees of different heights but with the same fitness. In the crown-shape game, each tree is assumed to choose the distribution of foliage along the height axis. A lone tree should have a hemispherical crown as the optimum. The monomorphic equilibrium in which all the trees in a community have the same crown shape is calculated. As the tree density increases, the average height of a crown of each tree increases and the height on the tree with the maximum foliage increases, but each tree should have some foliage all the way down to the ground. Several reasons for foliage cutoff at the lowest layer are discussed.