Forbidden subgraphs generating a finite set of graphs with minimum degree three and large girth

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 $n$.