  We study integral plane curves meeting at a single unibranch point and show
that such curves must satisfy two equivalent conditions. A numeric condition:
the local invariants of the curves at the contact point must be arithmetically
related. A geometric condition: the tropical curves that we associate to the
contact point must be isomorphic. Moreover, we prove closed formulas for the
delta-invariant of a unibranch singularity, and for the dimension of the loci
of curves with an assigned unibranch point. Our work is motivated by interest
in the Lang exceptional set.
