Invariance and attraction properties of Galton–Watson trees

We give a description of invariants and attractors the critical subcritical Galton–Watson tree measures under operation Horton pruning (cutting leaves with subsequent series reduction). Under regularity condition, class invariant consists binary one-parameter family trees offspring distribution {qk} that has power tail qk∼Ck−(1+1/q0), where q0∈(1/2,1). Each measure non-empty domain attraction consecutive pruning, specified by behavior initial distribution. The satisfy Toeplitz property for Tokunaga coefficients obey law exponent R=(1−q0)−1/q0.