We consider a canonical ensemble with a fixed number N of triangles for planar dynamical triangulation models with compact spin in the high temperature region. We find the asymptotics of the partition function Z(N) and reveal the analytic properties of the generating function U(x)=∑: Z(N)xN. New cluster expansion techniques are developed for this case. For fixed triangulation it would be quite standard but for random triangulations one has to deal with the non-zero entropy of the space between clusters. It is a multiscale expansion, where the role of scale is played by a topological parameter - the maximal length of chains of imbedded not simply connected clusters.