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 ZN(μ), 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.