  Given an arbitrary sequence of non-negative integers
$\vec{\lambda}=(\lambda_1,\dots,\lambda_n)$ and a graph $G$ with vertex set
$\{v_1,\dots,v_n\}$, the bounded degree complex, denoted
$\text{BD}^{\vec{\lambda}}(G)$, is a simplicial complex whose faces are the
subsets $H\subseteq E(G)$ such that for each $i \in \{1,\dots,n\}$, the degree
of vertex $v_i$ in the induced subgraph $G[H]$ is at most $\lambda_i$. When
$\lambda_i=k$ for all $i$, the bounded degree complex
$\text{BD}^{\vec{\lambda}}(G)$ is called the $k$-matching complex, denoted
$M_k(G)$.
  In this article, we determine the homotopy type of bounded degree complexes
of forests. In particular, we show that, for all $k\geq 1$, the $k$-matching
complexes of caterpillar graphs are either contractible or homotopy equivalent
to a wedge of spheres, thereby proving a conjecture of Julianne Vega
\cite[Conjecture 7.3]{Vega19}. We also give a closed form formula for the
homotopy type of the bounded degree complexes of those caterpillar graphs in
which every non-leaf vertex is adjacent to at least one leaf vertex.
