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Geodesic coalescence, or the tendency of geodesics to merge together, is ahallmark phenomenon observed in a variety of planar random geometries involvinga random distortion of the Euclidean metric. As a result of this, the union ofinteriors of all geodesics going to a fixed point tends to form a tree-likestructure which is supported on a vanishing fraction of the space. Suchgeodesic trees exhibit intricate fractal behaviour; for instance, while almostevery point in the space has only one geodesic going to the fixed point, thereexist atypical points which admit two such geodesics. In this paper, weconsider the directed landscape, the recently constructed scaling limit ofexponential last passage percolation (LPP), with the aim of developing tools toanalyse the fractal aspects of the tree of semi-infinite geodesics in a givendirection. We use the duality (Pimentel '16) between the geodesic tree and theinterleaving competition interfaces in exponential LPP to obtain a dualitybetween the geodesic tree and the corresponding dual tree in the landscape.Using this, we show that problems concerning the fractal behaviour of sets ofatypical points for the geodesic tree can be transformed into correspondingproblems for the dual tree, which might turn out to be easier. In particular,we use this method to show that the set of points admitting two semi-infinitegeodesics in a fixed direction a.s. has Hausdorff dimension $4/3$, therebyanswering a question posed in Busani-Sepp\"{a}l\"{a}inen-Sorensen '22. We alsoshow that the set of points admitting three semi-infinite geodesics in a fixeddirection is a.s. countable.

Duality in the directed landscape and its applications to fractal geometry