Given a system {S₁,…, S_N} of N contractive similarities satisfying some strong separation condition, it has an invariant set K for the system. In this
article, the authors construct some random measure $\mu_\o$ supported on
random subset $K_\o$ of $K$, $\mu_\o$ having some "non-standard" multifractal structure, which contrasts the well-known multifractal formalism for the invariant measure of system {S₁,…, S_N} may possess. The main tool is the multifractal structures of a Galton-Watson tree, which are obtained by Liu [9], Shieh--Taylor [14], and Morters--Shieh [12].