We discuss a set-theoretic characterization of trees that are Lindelöf with respect to the λ+ fine-wedge topology.

We then present a construction of a special λ+ Lindelöf tree of any successor-cardinal height from a suitable combinatorial axiom known as a proxy principle.

The construction is uniform and works equally well for successors of singular and regular cardinals.

We will focus on making the construction accessible to students, highlighting the "microscopic approach" to transfinite-tree construction

and the roles of the diamond and proxy principles.

This is joint work with Pedro Marun.