  A simple graph $G$ is an {\it 2-tree} if $G=K_3$, or $G$ has a vertex $v$ of
degree 2, whose neighbors are adjacent, and $G-v$ is an 2-tree. Clearly, if $G$
is an 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence
$\pi=(d_1,\ldots,d_n)$ of nonnegative integers is a {\it graphic sequence} if
it is realizable by a simple graph $G$ on $n$ vertices. Yin and Li (Acta
Mathematica Sinica, English Series, 25(2009)795--802) proved that if $k\ge 2$,
$n\ge \frac{9}{2}k^2+\frac{19}{2}k$ and $\pi=(d_1,\ldots,d_n)$ is a graphic
sequence with $\sum\limits_{i=1}^n d_i>(k-2)n$, then $\pi$ has a realization
containing every 1-tree (the usual tree) on $k$ vertices. Moreover, the lower
bound $(k-2)n$ is the best possible. This is a variation of a conjecture due to
Erd\H{o}s and S\'{o}s. In this paper, we investigate an analogue problem for
$2$-trees and prove that if $k\ge 3$ is an integer with $k\equiv i(\mbox{mod
}3)$, $n\geq20\lfloor\frac{k}{3}\rfloor^2+31\lfloor\frac{k}{3}\rfloor+12$ and
$\pi=(d_1,\ldots,d_n)$ is a graphic sequence with $\sum\limits_{i=1}^n
d_i>\max\{(k-1)(n-1),2\lfloor\frac{2k}{3}\rfloor
n-2n-\lfloor\frac{2k}{3}\rfloor^2+\lfloor\frac{2k}{3}\rfloor+1-(-1)^i\}$, then
$\pi$ has a realization containing every 2-tree on $k$ vertices. Moreover, the
lower bound $\max\{(k-1)(n-1),2\lfloor\frac{2k}{3}\rfloor
n-2n-\lfloor\frac{2k}{3}\rfloor^2+\lfloor\frac{2k}{3}\rfloor+1-(-1)^i\}$ is the
best possible. This result implies a conjecture due to Zeng and Yin (Discrete
Math. Theor. Comput. Sci., 17(3)(2016), 315--326).
