  The burning number is a recently introduced graph parameter indicating the
spreading speed of content in a graph through its edges. While the conjectured
upper bound on the necessary numbers of time steps until all vertices are
reached is proven for some specific graph classes it remains open for trees in
general. We present two different proofs for ordinary caterpillars and prove
the conjecture for a generalised version of caterpillars and for trees with a
sufficient amount of leaves. Furthermore, determining the burning number for
spider graphs, trees with maximum degree three and path-forests is known to be
$\mathcal{NP}$-complete, however, we show that the complexity is already
inherent in caterpillars with maximum degree three.
