  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 $h^\alpha$. 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
$\alpha$ of the height distribution function.

Probability