The $k$-in-a-tree problem for graphs of girth at least~$k$

For all integers $k\geq 3$, we give an $O(n^4)$ time algorithm for the problem whose instance is a graph $G$ of girth at least $k$ together with $k$ vertices and whose question is "Does $G$ contains an induced subgraph containing the $k$ vertices and isomorphic to a tree?". This directly follows for $k=3$ from the three-in-a-tree algorithm of Chudnovsky and Seymour and for $k=4$ from a result of Derhy, Picouleau and Trotignon. Here we solve the problem for $k\geq 5$. Our algorithm relies on a structural description of graphs of girth at least $k$ that do not contain an induced tree covering $k$ given vertices ($k\geq 5$).