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We explore the evolution of wave-function statistics on a finite Bethelattice (Cayley tree) from the central site ("root") to the boundary("leaves"). We show that the eigenfunction moments $P_q=N\left<|\psi|^{2q}(i)\right>$ exhibit a multifractal scaling $P_q\proptoN^{-\tau_q}$ with the volume (number of sites) $N$ at $N\to\infty$. Themultifractality spectrum $\tau_q$ depends on the strength of disorder and onthe parameter $s$ characterizing the position of the observation point $i$ onthe lattice. Specifically, $s= r/R$, where $r$ is the distance from theobservation point to the root, and $R$ is the "radius" of the lattice. Wedemonstrate that the exponents $\tau_q$ depend linearly on $s$ and determinethe evolution of the spectrum with increasing disorder, from delocalized to thelocalized phase. Analytical results are obtained for the $n$-orbital model with$n \gg 1$ that can be mapped onto a supersymmetric $\sigma$ model. Theseresults are supported by numerical simulations (exact diagonalization) of theconventional ($n=1$) Anderson tight-binding model.

Multifractality of wave functions on a Cayley tree: From root to leaves