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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 topicof signed degree sets and signed degree sequences has been studied from manydirections. In this paper, we study properties needed for signed trees to havea given signed degree set. We start by proving that $D$ is the signed degreeset of a tree if and only if $1\in D$ or $-1\in D$. Further, for every validset $D$, 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 orderthat a tree must have to realize $D$.

The Optimization of Signed Trees