It is well known that the height profile of a critical conditioned Galton–Watson tree with finite offspring variance converges, after a suitable normalisation, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of any order $\alpha<1$ , and that it is a.e. differentiable. We note that it cannot be differentiable at 0, but leave as open questions whether it is Lipschitz, and whether it is continuously differentiable on the half-line $(0,\infty)$ . The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest.

We have developed an easy-to-use, mouse-driven program for the interactive fitting of interstellar absorption lines in high-resolution astronomical spectra. The program, Xvoigt, gives values for the column density and velocity dispersion of the absorbing clouds. It runs under the popular X Window system available on most workstations, and offers significant enhancements over existing profile-fitting software. Xvoigt can be an important adjunct to automatic programs for fitting absorption lines in low to moderate signal-to-noise QSO or other spectra, and is ideal for demonstrating the details and difficulties of absorption line analysis.

Objectives: To assess the health and social functioning of patients attending a Child and Adolescent Mental Health Service (CAMHS) and to measure the impact of attendance using the Health of the Nation Outcome Scales for Children and Adolescents (HoNOSCA).

Method: HoNOSCA was completed on 73 consecutive patients attending for initial assessment with a review assessment being completed after six months or at discharge from the clinic if this occurred sooner on 53 of these. The impact of attendance at the clinics was determined by comparing initial and review mean HoNOSCA Scores.

Results: Of the 53, 66% were male and 34% female. Boys were more highly rated with regard to aggressive behaviour, performance in peer relationships and family life relationships whilst girls were rated as having more nonorganic and emotional symptoms. Older children showed the highest rates of poor school attendance, non accidental (self) injury and emotional problems while younger children showed the greatest aggressive behaviour and language skill problems. An improvement in the total HoNOSCA score from initial assessment to review was seen in 92%. There was an improvement in the HoNOSCA mean score from initial assessment to review.

Conclusions: Age, sex and symptom profiles of patients attending the service were similar to other CAMHS. Attendance at CAMHS produces improvements in patient outcomes over a six month period as measured using HoNOSCA, which proved to be a useful if somewhat time consuming tool.