LAUSR

We give a complete description of invariants and attractors of the critical and subcritical Galton-Watson tree measures under the operation of Horton pruning (cutting tree laves with subsequent series reduction). The class of invariant measures consists of the critical binary Galton-Watson tree and a one-parametric family of critical Galton-Watson trees with offspring distribution $\{q_k\}$ that has a power tail $q_k\sim k^{-(1+q_0)/q_0}$, where $q_0\in(1/2,1)$. Each invariant measure has a non-empty domain of attraction under consecutive Horton pruning, completely specified by the tail behavior of the initial Galton-Watson offspring distribution. The invariant measures are characterized by Toeplitz property of their Tokunaga coefficients; they satisfy Horton law with exponent $R = (1-q_0)^{-1/q_0}$.