Minor terminological disagreement with some coauthors: let T be a single-source shortest path tree in an unweighted undirected graph. Can you call T a "BFS tree"? Or is that only for trees that could be generated by breadth-first search?

E.g. consider shortest paths in \(K_{2,3}\) starting from the 3-vertex side. For breadth first search, two leaves from the 3-vertex side will share a parent. But there is a different shortest path tree where they have different parents. Is it a BFS tree?