We consider planar rooted random trees whose distribution is even for fixed height h and size N and whose height dependence is of exponential form e^−μh. Defining the total weight for such trees of fixed size to be Z^(μ)_N , we determine its asymptotic behaviour for large N, for arbitrary real values of μ. Based on this we identify the local limit of the corresponding probability measures and find a transition at μ = 0 from a single spine phase to a multi-spine phase. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for μ < 0 to the familiar quadratic growth at μ = 0 and to cubic growth for μ > 0.