A tree bijection for Eulerian orientations on quartic maps

In this paper we construct a recursive bijection between $4$-valent maps endowed with a Eulerian orientation – also known as the ice model – and a family of unary-binary trees. This answers a question of Bousquet-M'elou and Elvey Price, who established that both these families have the same generating function in arXiv:2503.15046 .