  We study nongeneric planar trees and prove the existence of a Gibbs measure
on infinite trees obtained as a weak limit of the finite volume measures. It is
shown that in the infinite volume limit there arises exactly one vertex of
infinite degree and the rest of the tree is distributed like a subcritical
Galton-Watson tree with mean offspring probability $m<1$. We calculate the rate
of divergence of the degree of the highest order vertex of finite trees in the
thermodynamic limit and show it goes like $(1-m)N$ where $N$ is the size of the
tree. These trees have infinite spectral dimension with probability one but the
spectral dimension calculated from the ensemble average of the generating
function for return probabilities is given by $2\beta -2$ if the weight $w_n$
of a vertex of degree $n$ is asymptotic to $n^{-\beta}$.
