Let $triangleleft$ be a relation between graphs. We say a graph $G$ is emph{$triangleleft$-ubiquitous} if whenever $Gamma$ is a graph with $nG triangleleft Gamma$ for all $n in mathbb{N}$, then one also has $aleph_0 G triangleleft Gamma$, where $alpha G$ is the disjoint union of $alpha$ many copies of $G$. The emph{Ubiquity Conjecture} of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation. In this paper, which is the first of a series of papers making progress towards the Ubiquity Conjecture, we show that all trees are ubiquitous with respect to the topological minor relation, irrespective of their cardinality. This answers a question of Andreae from 1979.
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