  A (Euclidean) greedy drawing of a graph is a drawing in which, for any two
vertices $s,t$ ($s \neq t$), 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. N\"{o}llenburg 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 $\leq 4$.
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.
