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 evaluate 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 $μ> 0$.