A finite one-sided tree of height h is defined as a rooted planar tree obtained by grafting branches on one side, say the right, of a spine, i.e. a linear path of length h starting at the root, such that the resulting tree has no simple path starting at the root of length greater than h. We consider the distribution on the set of one-sided trees T of fixed size N, such that the weight of T is proportional to , where is a real constant and h(T) denotes the height of T. We show that, for N large, has a weak limit as a probability measure supported on infinite one-sided trees. The dependence of the limit measure on shows a transition at from a single-spine phase for to a multi-spine phase for . Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for , to quadratic growth at , and to cubic growth for .