A (Euclidean) greedy drawing of a graph is a drawing in which, for any two vertices s, t (s = t), there is a neighbor vertex of s that is closer to t than to s in the Euclidean distance. Greedy drawings are important in the context of message routing in networks, and graph classes that admit greedy drawings have been actively studied. Nöllenburg and Prutkin (Discrete Comput. Geom., 58(3), pp. 543-579, 2017) gave a characterization of greedy-drawable trees in terms of an inequality system that contains a non-linear equation. Using the characterization, they gave a linear-time recognition algorithm for greedy-drawable trees of maximum degree ≤ 4. However, a combinatorial characterization of greedy-drawable trees of maximum degree 5 was left open. In this paper, we give a combinatorial characterization of greedy-drawable trees of maximum degree 5, which leads to a complete combinatorial characterization of greedy-drawable trees.Furthermore, we give a characterization of greedy-drawable pseudo-trees. It turns out that greedydrawable pseudo-trees admit a simpler characterization than greedy-drawable trees.