Random tree recursions: which fixed points correspond to tangible sets of trees?

Let $\mathcal{B}$ be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children $u$ and $v$ such that the subtrees rooted at $u$ and $v$ belong to it. Let $p$ be the probability that a Galton-Watson tree falls in $\mathcal{B}$. The metaproperty makes $p$ satisfy a fixed-point equation, which can have multiple solutions. One of these solutions is $p$, but what is the meaning of the others? In particular, are they probabilities of the Galton-Watson tree falling into other sets satisfying the same metaproperty? We create a framework for posing questions of this sort, and we classify solutions to fixed-point equations according to whether they admit probabilistic interpretations. Our proofs use spine decompositions of Galton-Watson trees and the analysis of Boolean functions.Comment: 41 pages; small changes in response to referees' comments; to appear in Random Structures & Algorithm

Independent advocacy in adult support and protection work

Purpose – The purpose of this paper is to consider independent advocates’ perspectives on their roles in Scottish adult support and protection (ASP) work, and the facilitators and barriers impacting on these roles in practice.  Design/methodology/approach – Semi-structured interviews were conducted with 20 managers and staff from six independent advocacy agencies operating across nine local authority areas.  Findings – Participants described key roles in supporting individuals to understand their rights and to negotiate ASP processes. They conceptualised their independence to be the key distinguishing feature of their role. Participants noted lower than expected rates of referral of ASP concerns to advocacy and variable experiences of communication with the statutory services. Particular emphasis was placed on the late stage at which many referrals are received. Awareness, understanding and acceptance of advocacy amongst the statutory services was felt to vary at both practice and strategic levels.  Research limitations/implications – The sample is not a representative one. However, some commonalities are worthy of note: particularly the participants’ commitment to ASP work and the perceived impact of statutory agencies on their involvement in it. The issue of late referrals merits some consideration at a national level. Issues of awareness and understanding amongst the statutory services, and their links with referral rates, are for further local-level exploration. The independent advocacy community might wish to discuss further the impacts on them of incorporation into statutory frameworks. Originality/value – Advocacy perspectives have been little drawn on in pre-existing ASP research