Let \({\mathcal F}_{n,p}\) be the subset of a simply generated family \(\mathcal F\) containing all trees \(T\) and \(p\) leaves and \(n\) nodes in total. Assuming that the trees are planar so that leaf numbers are identifiable, the paper deals with the following reliability model: (i) \(T\) is selected randomly from \({\mathcal F}_{n,p}\); (ii) Each edge of \(T\) has the same probability \(q\) of failing; (iii) Failures of edges occur independently.
The reliability variable \(Z_ i\) is defined as \(Z_ i= 1\) if the path connecting the root of \(T\) to the \(i\)th leaf does not contain a failed edge, 0 otherwise. In this article the stochastic process \(\{Z_ 1,\dots, Z_ p\}\) is studied. Asymptotic results when \(n\) and the path indices \(i\), \(j\) etc. tend to \(\infty\) are also obtained. The following auxiliary results are further derived: (i) Rotational invariance of leaf distance distribution in arbitrary simply generated families; (ii) The asymptotic height distribution of the \(i\)th leaf and the asymptotic joint height distribution of the \(i\)th and \(j\)th leaves in a random \(t\)-ary tree.