We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is given by a power function $h^α$. Defining the total weight for such trees of fixed size to be $Z_N$, a detailed analysis of the analyticity properties of the corresponding generating function is provided. Based on this, we determine the asymptotic form of $Z_N$ and show that the local limit at large size is identical to the Uniform Infinite Planar Tree, independent of the exponent $α$ of the height distribution function.