  A connected component of a vertex-coloured graph is said to be colourful if
all its vertices have different colours. By extension, a graph is colourful if
all its connected components are colourful. Given a vertex-coloured graph $G$
and an integer $p$, the Colourful Components problem asks whether there exist
at most $p$ edges whose removal makes $G$ colourful and the Colourful Partition
problem asks whether there exists a partition of $G$ into at most $p$ colourful
components. In order to refine our understanding of the complexity of the
problems on trees, we study both problems on $k$-caterpillars, which are trees
with a central path $P$ such that every vertex not in $P$ is within distance
$k$ from a vertex in $P$. We prove that Colourful Components and Colourful
Partition are NP-complete on $4$-caterpillars with maximum degree $3$,
$3$-caterpillars with maximum degree $4$ and $2$-caterpillars with maximum
degree $5$. On the other hand, we show that the problems are linear-time
solvable on $1$-caterpillars. Hence, our results imply two complexity
dichotomies on trees: Colourful Components and Colourful Partition are
linear-time solvable on trees with maximum degree $d$ if $d \leq 2$ (that is,
on paths), and NP-complete otherwise; Colourful Components and Colourful
Partition are linear-time solvable on $k$-caterpillars if $k \leq 1$, and
NP-complete otherwise. We leave three open cases which, if solved, would
provide a complexity dichotomy for both problems on $k$-caterpillars, for every
non-negative integer $k$, with respect to the maximum degree. We also show that
Colourful Components is NP-complete on $5$-coloured planar graphs with maximum
degree $4$ and on $12$-coloured planar graphs with maximum degree $3$. Our
results answer two open questions of Bulteau et al. mentioned in [30th Annual
Symposium on Combinatorial Pattern Matching, 2019].
