  For a family $\mathcal{H}$ of graphs, a graph $G$ is said to be {\it
$\mathcal{H}$-free} if $G$ contains no member of $\mathcal{H}$ as an induced
subgraph. We let $\tilde{\mathcal{G}}_{3}(\mathcal{H})$ denote the family of
connected $\mathcal{H}$-free graphs having minimum degree at least $3$. In this
paper, we characterize the non-caterpillar trees $T$ having diameter at least
$7$ such that $\tilde{\mathcal{G}}_{3}(\{C_{3},C_{4},T\})$ is a finite family,
where $C_{n}$ is a cycle of order $n$.
