 Solution

Assess the observation-error assumption with standardized residual plots and quantile-quantile plots, preferably comparing the same discrepancies with posterior predictive replicated data. Inspect tail behavior, asymmetry, variance versus fitted size and serial patterns within a tree; fitted residuals are dependent and do not behave exactly like independent errors.

This does not test normality of the tree random effects. For that, simulate new tree-specific log asymptotes from the population hierarchy and compare between-tree discrepancies or use leave-one-tree-out predictions. There are only five trees, so distributional checks on their population are intrinsically weak. Conditioning on the five fitted effects can hide misspecification of their normal population law.