Tropical compactification and the Gromov--Witten theory of\n $\\mathbb{P}^1$

We use tropical and nonarchimedean geometry to study the moduli space of\ngenus $0$ stable maps to $\\mathbb{P}^1$ relative to two points. This space is\nexhibited as a tropical compactification in a toric variety. Moreover, the fan\nof this toric variety may be interpreted as a moduli space for tropical\nrelative stable maps with the same discrete data. As a consequence, we confirm\nan expectation of Bertram and the first two authors, that the tropical Hurwitz\ncycles are tropicalizations of classical Hurwitz cycles. As a second\napplication, we obtain a full descendant correspondence for genus $0$ relative\ninvariants of $\\mathbb{P}^1$.\n