A quasi order \(Q\) is called a \(\Delta^0_2\)-better quasi order if every \(\Delta^0_2\) array into \(Q\) is good. This notion plays an important role in Montalb'an’s proof of Fra"iss'e’s conjecture, originally proved by Laver, in the system \(\Pi^1_1\)-CA\(_0\). We show that the property of being \(\Delta^0_2\)-better quasi order can be characterized in terms of an appropriate class of ill founded labelled trees, and discuss that result in the context of reverse mathematics.