We investigate the tree gonality of a genus-g metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this number is at most $\lceil g/2\rceil +1$, a fact whose proofs so far required an algebro-geometric detour via special divisors on curves. For even genus, the tropical morphism which realizes the bound belongs to a family of tropical morphisms that is pure of dimension $3g-3$ and that has a generically finite-to-one map onto the moduli space of genus-g metric graphs. Our methods focus on the study of such families. This is part I in a series of two papers: in part I we fix the combinatorial type of the metric graph to show a bound on tree-gonality, while in part II we vary the combinatorial type and show that the number of tropical morphisms, counted with suitable multiplicities, is the same Catalan number that counts morphisms from a general genus-g curve to the projective line.

我们研究了树gonality一个属─摹指标图形，定义为从测量图中的任何修改热带热带射的最小程度的指标树。我们给出了一个组合的建设性证明，即这个数字最多$\lceil G/2\rceil +\mathrm{1个}$，到目前为止，要证明这一事实，需要通过曲线上的特殊除数来进行代数几何变换。对于偶属来说，实现界的热带态射属于一个纯粹的维态族$3G-3$和具有一般有限一对一映射到的属─模空间克度量的曲线图。我们的方法着重研究这类家庭。这是两篇论文系列的第一部分：在第一部分中，我们固定了度量图的组合类型以显示对树的限制，而在第二部分中，我们改变了组合类型并显示了热带变态的数量，与合适的多重计数，是相同的Catalan数，从一个普通的属─计数态射克曲线投影线。