  A signed graph $G$ is a graph where each edge is assigned a + (positive edge)
or a - (negative edge). The signed degree of a vertex $v$ in a signed graph,
denoted by $sdeg(v)$, is the number of positive edges incident to $v$
subtracted by the number of negative edges incident to $v$. Finally, we say $G$
realizes the set $D$ if: $$ D = \{sdeg(v) \text{ : } v\in V(G) \}. $$ The topic
of signed degree sets and signed degree sequences has been studied from many
directions. In this paper, we study properties needed for signed trees to have
a given signed degree set. We start by proving that $D$ is the signed degree
set of a tree if and only if $1\in D$ or $-1\in D$. Further, for every valid
set $D$, we find the smallest diameter that a tree must have to realize $D$.
Lastly, for valid sets $D$ with nonnegative numbers, we find the smallest order
that a tree must have to realize $D$.
