  Given integers $n\ge \Delta\ge 2$, let $\mathcal{T}(n, \Delta)$ be the
collection of all $n$-vertex trees with maximum degree at most $\Delta$. A
question of Alon, Krivelevich and Sudakov in 2007 asks for determining the best
possible spectral gap condition forcing an $(n, d,\lambda)$-graph to be
$\mathcal{T}(n, \Delta)$-universal, namely, it contains all members of
$\mathcal{T}(n, \Delta)$ as a subgraph simultaneously. In this paper we show
that for sufficiently large integer $n$ and all $\Delta\in \mathbb{N}$, every
$(n, d,\lambda)$-graph with \[ \lambda\le\frac{d}{2\Delta^{5\sqrt{\log n}}} \]
is $\mathcal{T}(n, \Delta)$-universal. As an immediate corollary, this implies
that Alon's ingenious construction of triangle-free sparse expander is
$\mathcal{T}(n, \Delta)$-universal, which provides an explicit construction of
such graphs and thus solves a question of Johannsen, Krivelevich and Samotij.
Our main result is formulated under a much more general context, namely, the
$(n,d)$-expanders. More precisely, we show that there exist absolute constants
$C,c>0$ such that the following statement holds for sufficiently large integer
$n$.
  (1).For all $\Delta\in \mathbb{N}$, every $(n, \Delta^{5\sqrt{\log
n}})$-expander is $\mathcal{T}(n, \Delta)$-universal.
  (2).For all $\Delta\in \mathbb{N}$ with $\Delta \le c\sqrt{n}$, every $(n,
C\Delta n^{1/2})$-expander is $\mathcal{T}(n, \Delta)$-universal.
  Both results significantly improve a result of Johannsen, Krivelevich and
Samotij, and have further implications in locally sparse expanders and
Maker-Breaker games that also improve previously known results drastically.
