Invariance and attraction properties of Galton–Watson trees

Abstract

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 {qk} that has a power tail qk∼Ck−(1+1/q0), where q0∈(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−q0)−1/q0.