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We consider the i.i.d. Bernoulli field $\mu_p$ with occupation density $p \in(0,1)$ on a possibly non-regular countably infinite tree with bounded degrees.For large $p$, we show that the quasilocal Gibbs property, i.e. compatibilitywith a suitable quasilocal specification, is lost under the deterministictransformation which removes all isolated ones and replaces them by zeros,while a quasilocal specification does exist at small $p$. Our results providean example for an independent field in a spatially non-homogeneous setup whichloses the quasilocal Gibbs property under a local deterministic transformation.

Gibbs Properties of the Bernoulli field on inhomogeneous trees under the removal of isolated sites