  An antimagic labeling for a graph $G$ with $m$ edges is a bijection $f: E(G)
\to \{1, 2, \dots, m\}$ so that $\phi_f(u) \neq \phi_f(v)$ holds for any pair
of distinct vertices $u, v \in V(G)$, where $\phi_f(x) = \sum_{x \in e} f(e)$.
A strongly antimagic labeling is an antimagic labeling with an additional
condition: For any $u, v \in V(G)$, if $\deg(u) > \deg(v)$, then $\phi_f(u) >
\phi_f(v)$. A graph $G$ is strongly antimagic if it admits a strongly antimagic
labeling. We present inductive properties of strongly antimagic labelings of
graphs. This approach leads to simplified proofs that spiders and double
spiders are strongly antimagic, previously shown by Shang [Spiders are
antimagic, Ars Combinatoria, 118 (2015), 367--372] and Huang [Antimagic
labeling on spiders, Master's Thesis, Department of Mathematics, National
Taiwan University, 2015], and by Chang, Chin, Li and Pan [The strongly
antimagic labelings of double spiders, Indian J. Discrete Math. 6 (2020),
43--68], respectively. We fix a subtle error in [The strongly antimagic
labelings of double spiders, Indian J. Discrete Math. 6 (2020), 43--68].
Further, we prove certain level-wise regular trees, cycle spiders and cycle
double spiders are all strongly antimagic.
