S.V.Chmutov, S.V.Duzhin, S.K.Lando, Vassiliev knot invariants} II. Intersection graph conjecture for trees, The principal tool in this sequence of papers is the intersection graph of a chord diagram: this is obtained by taking a vertex for each chord in the diagram and an edge between two vertices if the corresponding chords intersect. The intersection graph conjecture is that weight systems (functionals on chord diagrams satisfying the 4T and 1T relations) depend only on the intersection graphs of diagrams. This has been verified for weight systems up to degree eight and for weight systems coming from the defining representations of $so(n)$ and $gl(n)$. This paper (second in the series) consists of proving the conjecture for the case that the intersection graph is a tree.