A stability theorem for embedding bounded degree spanning trees

We prove that if an $n$-vertex graph $G$ is non-extremal and $T$ is a bounded degree tree on $n$ vertices, then $T\subset G$ even when the minimum degree of $G$ is less than $n/2$ by a linear term. We avoid the use of the Regularity lemma, instead we apply a vertex decomposition theorem by the author, which does not require a tower-type lower bound for $n.$