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