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.